Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2013 Article ID 432192 10 pageshttpdxdoiorg1011552013432192

Research ArticleA Sixth Order Accuracy Solution to a System of NonlinearDifferential Equations with Coupled Compact Method

Don Liu1 Qin Chen2 and Yifan Wang1

1 Mathematics and Statistics Louisiana Tech University Ruston LA 71272 USA2Civil and Environmental Engineering Louisiana State University Baton Rouge LA 70803 USA

Correspondence should be addressed to Don Liu donliulatechedu

Received 26 June 2013 Accepted 4 October 2013

Academic Editor Marek Krawczuk

Copyright copy 2013 Don Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations Through a special formulation a system of nonlinear partial differential equations was solved alternately andexplicitly in time without linearizing the nonlinearityCoupled compact schemes of sixth order accuracy in space were developed toobtain numerical solutionsWithin couple compact schemes variables and their first and second derivatives were solved altogetherThe sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system isbanded instead of blocked This facilitates solving very large systems The efficiency simplicity and accuracy make this coupledcompact method viable as variational and weighted residual methods Results were compared with exact solutions which wereobtained via devised forcing terms Error analyses were carried out and the sixth order convergence in space and second orderconvergence in time were demonstrated Long time integration was also studied to show stability and error convergence rates

1 Introduction

Compact differencemethods essentially the implicit versionsof finite different methods are superior to the explicit ver-sions in achieving high order accuracy High order compactmethods directly approximate all derivatives with high accu-racy without any variational operation or projection Theyare very effective for complex and strongly nonlinear partialdifferential equations especially when variational methodsare inconvenient to implement They have smaller stencilsthan explicit finite difference methods while maintaininghigh order accuracy

Much development has been achieved in finite differencemethods A large number of papers related to various finitedifference algorithms have been published over the pastdecades Kreiss [1] was among the first who pioneered theresearch on compact implicit methods Hirsh [2] developedand applied a 4th order compact scheme to Burgersrsquo equationin fluid mechanics Adam [3] developed 4th order compactschemes for parabolic equations on uniform grids Hoffman[4] studied the truncation errors of the centered finitedifference method on both uniform and nonuniform grids

and discussed the accuracy of these schemes after applyinggrid transformations Dennis and Hudson [5] studied a 4thorder compact scheme to approximate Navier-Stokes typeoperators Rai andMoin [6] presented several finite differencesolutions for an incompressible turbulence channel flowThey also compared their solutions of high order upwindschemes with solutions obtained from spectral methodsTheir work demonstrated that finite difference methods canbe a good approach for the direct numerical simulation ofturbulence Lele [7] demonstrated the spectral-like resolutionof the classical Pade schemes and the flexibility of compactfinite difference methods He developed compact differencemethods for mid-point interpolation and filtering with highorder formal accuracy Yu et al [8] solved an unsteady 2DEuler equation with a 6th order compact difference schemein space and a 4th order Runge-Kutta scheme in time whichprovide a good example of high order (both spatial andtemporal) finite difference methods Mahesh [9] presentedhigh order compact schemes with both the first and secondorder derivatives evaluated simultaneously Dai and Nassar[10 11] developed high order compact difference schemes

2 Journal of Computational Engineering

for high order derivatives in time and space in solving a 3Dheat transport problem at the microscale Sun and Zhang[12] proposed a new high order finite difference discretizaionstrategy based on the Richardson extrapolation techniqueand on an operator interpolation scheme for solving one- andtwo-dimensional convection-diffusion equations

Compactmethods arewidely used in applications in engi-neering and physics Ekaterinaris [13] presents fourth orderfactorized upwind compact schemes for time accurate solu-tions of hyperbolic equation systems Li et al [14] obtainedsolutions of Navier-Stokes equation expressed in the streamfunction-vorticity form with a fourth order compact schemefor a driven cavity problem Shang [15] solved the time-dependent Maxwell equation with spurious high-frequencyoscillatory components of the numerical solution beingsuppressed by a spatial filter Spotz and Carey extended high-order compact schemes to time-dependent problems [16]El-Zoheiry [17] devised a three-level iterative scheme basedon the compact implicit method for solving the improvedBoussinesq equation Zhang et al [18] derived a fourthorder compact scheme with multigrid mesh refinement forconvection diffusion problems Central and upwind compactmethods were analyzed by Sengupta et al [19] and schemeswith improved boundary closure and an explicit high-orderupwind schemewere proposed Shen et al [20] demonstrateda new way of constructing high accuracy shock-capturingcompact schemes Jocksch et al [21] developed a fifth ordercompact upwind-biased finite difference scheme for com-pressible Couette flow Mohamad et al [22] presented a thirdorder compact upwind scheme for flows containing disconti-nuities Anthonio and Hall [23] simulated free surface flowswith a high-order compact finite difference method and afourth order Runge-Kutta method for temporal discretiza-tion Cruz and Chen [24] modeled porous beds with highorder time-marching fourth order finite difference schemesA dual-compact scheme with the increased dispersive accu-racy for the convective terms was proposed byChiu and Sheu[25] A new family of high-order compact upwind differenceschemes with good spectral resolution was introduced byZhou et al [26] Sanyasiraju andMishra [27] developed sixthorder compact schemes for convection diffusion equationswith constant coefficients Dehghan and Mohebbi [28] pre-sented unconditionally stable fourth order accuracy in bothspace and time solution to unsteady convection diffusionequation Shah et al [29] solved the Navier-Stokes equationsin which the convective terms were approximated with athird order upwind compact scheme using flux differencingsplitting Xie et al [30] presented the numerical solution ofthe one dimensional nonlinear Schrodinger equation Tonelliand Petti [31] used a finite volume technique to solve theadvective part of the Boussinesq equations and applied a finitedifference method to discretize dispersive and source termsThe time integration was performed with the fourth orderAdams-Bashforth-Moulton method

Irregular geometries could hinder the application ofcompact methods To facilitate nonuniform discretizationone way is to use a transformation so that compact methodscan be implemented in the transformed space where thenominal rate of accuracy could be achieved For example

zeros of Chebyshev polynomials could be adopted as the gridpoints for computation [32 33] Gamet et al [34] developeda 4th order compact scheme for approximating the firstorder derivative with a full inclusion of metrics directly ona nonuniform mesh with variable grid spacing Using thismethod a direct numerical simulation of a compressible tur-bulent channel flow was conducted Vasilyev [35] presentedhigh order finite difference simulation of turbulent flowson nonuniform meshes with good conservation propertiesHermanns and Hernandez [36] developed high order finitedifference methods on nonuniform grids Usually the actualaccuracy is determined by the effect of the transformationAnother option for nonuniform grids is to use full inclusionof metrics (FIM) such as Liu et al [33] where fourth andsixth order compact methods were used for parabolic partialdifferential equations encountered in geophysical modelingGenerally speaking FIM could be difficult for deriving highaccuracy schemes

In the past two decades Boussinesq-type equationsderived fromdepth-integration of conservation laws formassandmomentum have beenwidely used to simulate nonlinearwater waves in coastal and ocean engineering One of thechallenges of solving this class of partial differential equationsis the numerical treatment of the dispersive terms describedby the mixed third order derivative in space (twice) and time

One objective of this paper is to develop 6th ordercoupled compact schemes formodified Boussinesq equationswith analytical exact solution for comparisonThese schemesdemonstrate high order accuracy and fast convergence ratesin both space and time The idea in this paper could be usedfor solving other systems of coupled nonlinear equations

This paper is outlined as below In simulation approachesthe modified Boussinesq equations and their exact solutionscorresponding to specially devised forcing functions aregiven Then the solution ideas and procedures are presentedfollowed by schemes of sixth order accuracy developed forthese equations In simulation results the solutions of all cou-pled variables and the convergence rates in space and time aredemonstrated In addition the effect of long time integrationis investigated to show numerical stability over time Finallyconclusion summarises the novelty and uniqueness of theschemes developed in this paper and future work is planned

2 Simulation Approaches

21 Modified Boussinesq Equations In terms of the velocity119906 and water depth 120578 Boussinesq-type equations in onehorizontal dimension could bemodified into below equationswhich are similar to [37]

120597120578

120597119905

+

120597 (119906120578)

120597119909

= 119891 (119906 120578 119909 119905)

120597119906

120597119905

+ 119906

120597119906

120597119909

minus

120583

3

1205973119906

1205971199092120597119905

+

120597120578

120597119909

= 119892 (119906 120578 119909 119905)

(1)

where 120583 is a dimensionless parameter measuring the wavedispersion at a given still-water depth The above equationsare closed with two initial conditions one for 119906 and one for120578 and boundary conditions two for 119906 and one for 120578 This is

Journal of Computational Engineering 3

a system of nonlinearly coupled partial differential equationswith strong convection and dispersivity

To solve this system of equations by the method of lines[38] (1) could be formulated into a system of time-dependentequations in terms of 120578 and a new variable 119908

120597120578

120597119905

= 119865

120597119908

120597119905

= 119866

(2)

where

119865 = 119891 (119906 120578 119909 119905) minus

120597 (119906120578)

120597119909

= 119891 (119906 120578 119909 119905) minus 120578

120597119906

120597119909

minus 119906

120597120578

120597119909

(3)

119908 = 119906 minus

120583

3

1205972119906

1205971199092 119866 = 119892 (119906 120578 119909 119905) minus 119906

120597119906

120597119909

minus

120597120578

120597119909

(4)

Nowwe could take care of the systemof equations in time andspace separately To march (2) in time a variety of methodsare available [32 38 39] Since implicit methods are used inspace and the system is strongly nonlinear we select explicitmethods for time evolution In this paper up to third orderAdams-Bashforth methods are used to integrate in timeDuring the initial time steps lower order Adams-Bashforthmethods are used instead

In order to compare results and investigate the rate ofconvergence exact solutions to (1) are needed With somespecific forms of 119891 and 119892 in these equations with 120583 = 3 theexact solutions to (1) are available In this paper we presentthe special forms of 119891 and 119892 and the corresponding exactsolutions as given below

119891 = (2119909 + 1) 1198902119909minus2119905

minus 119890119909minus119905

119892 = (1199092+ 119909) 119890

2119909minus2119905+ 3119890119909minus119905

(5)

subject to the initial conditions

120578 (119909 0) = 119890119909 119906 (119909 0) = 119909119890

119909 (6)

and the boundary conditions

120578 (1 119905) = 1198901minus119905 119906 (1 119905) = 119890

1minus119905

120597119906 (2 119905)

120597119909

= 31198902minus119905

(7)

the exact solutions to (1) are

120578 = 119890119909minus119905

119906 = 119909119890119909minus119905

1 le 119909 le 2 119905 ge 0 (8)

22 Solution Ideas and Procedures Themajor steps in findingnumerical solution are elaborated below These steps involveCompact Schemes I II and III which are described in the nextsubsection We use superscripts to denote the time levels ofvariables

Based on the initial 119906(0) and 120578(0) we use Compact SchemeI to compute 120597119906

(0)120597119909 and 120597120578

(0)120597119909 then by (3) and (4)

to acquire 119865(0) and 119866(0) and we use Compact Scheme II to

compute 1205972119906(0)1205971199092 from 119906

(0) then by (4) to obtain 119908(0)

With 119865(0) 119866(0) 120578(0) and 119908(0) being known by the first orderAdams-Bashforth method 120578(1) and 119908(1) could be computedNext we use Compact Scheme III to compute 119906

(1) Thisconcludes the computation in time level one

Based on 119906(1) and 120578

(1) we use Compact Scheme Ito compute 120597119906(1)120597119909 and 120597120578

(1)120597119909 then by (3) and (4) to

acquire 119865(1) and 119866(1) With 119865(1) 119866(1) 119865(0) 119866(0) 120578(1) and 119908(1)being known by the second order Adams-Bashforthmethod120578(2) and 119908

(2) could be computed Next we use CompactScheme III to compute 119906(2) This finishes the computation intime level two

Based on 119906(2) and 120578

(2) repeat the same procedures anduse the third order Adams-Bashforthmethod to compute 120578(3)

and 119906(3) Continue the computations and use the third orderAdams-Bashforth method from now on to compute 119906 and 120578at the next time level

23 Coupled Compact Schemes of High Accuracy Now wefocus on the spatial discretization and the specific high ordercompact schemesmentioned earlierWepartition the domainof interest 1 le 119909 le 2 into119873 equal intervals with119873+1 nodeswith the nodal index starting from 0 to the last one119873

231 Compact Scheme I Solving for 120597119906120597119909120597120578120597119909withKnown119906 120578 Denote either 119906 or 120578 with 119884 The first derivative of 119884relative to 119909 at node 119909

119894is 1198841015840119894= 120597119884119894120597119909 The uniform grid

size is ℎ Based on the known values of 119906 and 120578 at the currenttime level (eg the initial values) 120597119906120597119909 and 120597120578120597119909 could becomputed accurately with compact methods For the interiornodes the scheme of the sixth order accuracy is

121198841015840

119894minus1+ 36119884

1015840

119894+ 12119884

1015840

119894+1

= ℎminus1(minus119884119894minus2

minus 28119884119894minus1

+ 28119884119894+1

+ 119884119894+2) + 119900 (ℎ

6)

(9)

For the two left end nodes use the sixth order scheme with119894 = 1 2 respectively

minus601198841015840

119894minus1minus 300119884

1015840

119894= ℎminus1(197119884

119894minus1+ 25119884

119894minus 300119884

119894+1

+100119884119894+2

minus 25119884119894+3

+ 3119884119894+4)

+ 119900 (ℎ6)

(10)

For the two right end nodes when solving for 120597120578120597119909 use thesixth order scheme below with 119894 = 119873119873 minus 1 respectivelywhen solving for 120597119906120597119909 use the same scheme with 119894 = 119873 minus

1 and the Neumann boundary condition

minus 601198841015840

119894minus 300119884

1015840

119894+1

= ℎminus1(minus197119884

119894minus 25119884

119894minus1+ 300119884

119894minus2minus 100119884

119894minus3

+25119884119894minus4

minus 3119884119894minus5) + 119900 (ℎ

6)

(11)

Equations (9) (10) and (11) form a tridiagonal system ofdimension119873+ 1 by119873+ 1 in terms of the nodal values of the

4 Journal of Computational Engineering

first derivative 1198841015840 The coefficient matrix of the linear systemis stored as three vectors in order to save storage The linearsystem is solved with theThomas Algorithm [40]

232 Compact Scheme II Solving for 12059721199061205971199092 with Known 119906Denote 119906 with 119884 11988410158401015840

119894= 12059721198841198941205971199092 Based on the known values

of 119906 at the current time level (eg the initial time) 12059721199061205971199092could be computed with high order compact methods Forthe interior nodes the scheme of the sixth order accuracy is

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1

= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(12)

For the two left end nodes use the following sixth orderscheme with 119894 = 1 2 respectively with the Dirichletboundary condition 119884

0= 1198901minus119905 at 119909 = 1 included

12057211988410158401015840

119894minus1+ 12057311988410158401015840

119894= ℎminus2(119887119884119894minus1

+ 1198860119884119894+ 1198861119884119894+1

+ 1198862119884119894+2

+ 1198863119884119894+3

+ 1198864119884119894+4

+1198865119884119894+5) + 119900 (ℎ

6)

(13)

where 120572 = 2823(13) 120573 = 32340 119887 = 37350(1927) 1198860=

minus75537 1198861= 36767(12) 119886

2= 4762(1627) 119886

3= minus4620

1198864= 1463 and 119886

5= (1154) minus 187 For the right end node

use the fifth order mixed scheme as below with the Neumannboundary condition 1198841015840

119873= 31198902minus119905 at 119909 = 2 included

7211988410158401015840

119873minus1+ 18119884

10158401015840

119873minus2= minus 30ℎ

minus11198841015840

119873

+ ℎminus2(127119884

119873minus 216119884

119873minus1

+81119884119873minus2

+ 8119884119873minus3

) + 119900 (ℎ5)

(14)

For the node next to the right end node use the sixth orderscheme below with 119894 = 119873 and the same coefficients as in (13)

12057311988410158401015840

119894minus1+ 12057211988410158401015840

119894= ℎminus2(1198865119884119894minus6

+ 1198864119884119894minus5

+ 1198863119884119894minus4

+ 1198862119884119894minus3

+ 1198861119884119894minus2

+ 1198860119884119894minus1

+119887119884119894) + 119900 (ℎ

6)

(15)

Equations (12) (13) (14) and (15) form a quasitridiagonalsystem of dimension 119873 + 1 by 119873 + 1 in terms of the nodalvalues of the second derivative of 119906 since only one nonzeroelement in the last row lies outside the tridiagonal lines Thecoefficient matrix of the linear system could still be storedas three vectors The linear system is solved by the ThomasAlgorithm with a minor modification

233 Compact Scheme III Solving for 119906 12059721199061205971199092 with Known119908 After the time integration with the Adams-Bashforthmethod 119908 at the next time level becomes known Now we

compute the new nodal values of 119906 and 12059721199061205971199092 at the nexttime level with 119908 based on (4)

119906 minus

1205972119906

1205971199092= 119908 (16)

Since two variables are solved for at the same time anotherequation is needed here A compact scheme of the sixth orderaccuracy as below could be the supplementary equation

sum

119894

119886119894

1205972119906119894

1205971199092=

1

ℎ2sum

119896

119887119896119906119896+ 119900 (ℎ

6) (17)

Therefore the resulting system is of dimension 2119873 + 2 by2119873+2 in terms of the nodal values of 12059721199061205971199092 and 119906 In otherwords there are two equations for each node Denote 119906 with119884 at the node 119894 119906

119894= 119884119894 12059721199061198941205971199092= 12059721198841198941205971199092= 11988410158401015840

119894 For the

interior nodes we choose (17) to be (12) Then the system of(16) and (17) at the interior nodes becomes

minus11988410158401015840

119894+ 119884119894= 119908119894 (18)

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(19)

For the node next to the very end node on each side (18) with119894 = 1 for the left and 119894 = 119873minus1 for the right is implementedThesecond equation at the same node is the sixth order schemeas below with 119894 = 3 for the left one and 119894 = 119873minus1 for the rightone

11988410158401015840

119894minus3minus 24119884

10158401015840

119894minus2minus 194119884

10158401015840

119894minus1minus 24119884

10158401015840

119894+ 11988410158401015840

119894+1

= ℎminus2(minus240119884

119894minus2+ 480119884

119894minus1minus 240119884

119894) + 119900 (ℎ

6)

(20)

For the left end node at 119909 = 1 the Dirichlet boundarycondition for 119884 is implemented The second equation at thesame node is (13) with 119894 = 1 and the Dirichlet boundarycondition appended For the right end node at 119909 = 2 (18)is implemented with 119894 = 119873 The second equation at the samenode is the fifth order mixed scheme that is (14) with 119894 = 119873

and the Neumann boundary condition1198841015840119873= 31198902minus119905 appended

We specially arranged 11988410158401015840

119894and 119884

119894alternately that is 11988410158401015840

0

11988401198841015840101584011198841 to acquire a banded linear system of dimension

2119873 + 2 by 2119873 + 2 in terms of the nodal values of 12059721199061205971199092and 119906 Without such deliberate alternating arrangement thecoefficientmatrixwould be blockedwith sparsityThis systemwith a small bandwidth could be efficiently solved via thebanded Gaussian elimination with scaled partial pivoting

3 Simulation Results

31 Approximate and Exact Solutions We compare ourapproximate (numerical) solutions with exact solutionsfor 119906 and its derivatives in Figure 1(a) 120578 and its derivativein Figure 1(b) at the dimensionless time 119879 = 01 The timestep size is 119889119905 = 00001 and the space step size is 119889119909 = 005No visual difference could be seen from these figures betweenthe approximate and exact solutions To investigate errors weneed to check convergence rates in both space and time

Journal of Computational Engineering 5

x

Valu

e

1 125 15 175 2

4

8

12

16

20

24

28

ApproxuApproxuxApprox uxx

Exact uExact uxExact uxx

(a)

x

1 125 15 175 2

+ + + + ++

++

++

++

++

++

++

++

+

Valu

e

3

4

5

6

7

+Approx120578Exact 120578

Approx120578xExact 120578x

(b)

Figure 1 Comparison of the approximate and exact solutions for 119906 (a) and 120578 (b)

50 100 150 200

Conv rate

10minus13

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

N

ux = 623

120578x

ux

120578x

= 612

(a)

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

L2

(rel

ativ

e err

or)

50 100 150 200N

Conv rate uxx = 593

(b)

Figure 2 Convergence rates for Compact Scheme I (a) and Compact Scheme II (b) in space

32 Convergence Rates of Separate Compact Schemes in Space

321 Compact Scheme I The performance of CompactScheme I is examined here Since the exact solutions areavailable we compute the Euclidean (119871

2) norms of the

relative errors for the first derivative of 119906 and 120578 at all nodesFigure 2(a) presents in the log-log scale the 119871

2norm of

the relative errors versus the number of nodes used indiscretizationWhen the spatial resolution goes down to 001that is 119873 = 100 the Euclidean norms of the relative errors

6 Journal of Computational Engineering

50 100 150 200N

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4L2

(rel

ativ

e err

or)

Conv rateu = 557

uxx = 610

(a)

50 100 150 200N

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

Conv rateu = 630

ux = 581

uxx = 484

120578 = 682

120578x = 537

dt = 10minus4

(b)

Figure 3 (a) Convergence rates for Compact Scheme III (b) Convergence rates for combined schemes in space with 119889119905 = 10minus4

drop to themachine zero in double precisionThe data pointsdemonstrate the sixth order convergence rate for both 120597119906120597119909and 120597120578119889119909

322 Compact Scheme II To see the performance of Com-pact Scheme II we use the known exact solution to computethe Euclidean norm of the relative errors for 12059721199061205971199092 at allnodes is In the log-log scale Figure 2(b) shows the 119871

2norm

of the relative error versus the number of nodes used in spaceBefore the spatial step reduces to 001 the 119871

2norm of the

relative errors of 12059721199061205971199092 drops to the machine zero Thisfigure clearly demonstrates the sixth order convergence rate

323 Compact Scheme III The performance of CompactScheme III is demonstrated below Taking advantage of theknown exact solution we compute the Euclidean normsof the relative errors for both 119906 and 120597

21199061205971199092 at all nodes

Figure 3(a) plots the 1198712norms of the relative errors versus

the number of nodes used in discretization The linearsystem generated from the Compact Scheme III is no longertridiagonal but it is still banded Because a fifth order schemeis used at the right end node 119894 = 119873 in order to adopt theNeumann boundary condition the convergence rates for 119906and 12059721199061205971199092 are slightly different but about sixth order

33 Convergence Rates of Combined Schemes in Space

331 Time Step 119889119905 = 10minus4 Nowwe examine the convergence

rates for all three compact schemes together over 1000 timesteps after the initial conditions are imposed During thisperiod of time Compact Schemes I II and III are used one

after another to solve these modified Boussinesq equationsFigure 3(b) presents the overall spatial convergence rates(indicated in the legend resp) for all variables after 1000 timesteps which was set to be 119889119905 = 10

minus4 The lowest convergencerate of 484 is observed for 12059721199061205971199092 because the variable 119906carries a second derivative and the boundary condition is notexplicitly given in terms of 12059721199061205971199092 The rest of the variableshave the sixth order convergence rate

332 Time Step 119889119905 = 10minus5 and 119889119905 = 10

minus6 Similar to Figures3(b) and 4(a) with 119889119905 = 10

minus5 and Figure 4(b) with 119889119905 = 10minus6

present the spatial convergence rates (indicated in the legend)involving Compact Schemes I II and III for all variables after1000 time steps The lowest convergence rate about 49 isexpected for 12059721199061205971199092 for the same reason as before For therest variables the sixth order convergence rates are clearlydemonstrated From Figures 3(b) 4(a) and 4(b) we observethat as the time step size reduces high order convergencerates are maintained for higher spatial resolution (larger119873 infigures)This is because a smaller time step size produces lesserror during time integration However as the spatial resolu-tion (119873) increases the rounding error increases accordinglyand eventually overtakes the truncation error Therefore wedo not see the errors keep decreasing as119873 increases

34 Convergence Rate in Time Integration After we exam-ined the spatial convergence rates now we demonstrate theconvergence rate in time integration To this end we fix thespatial step size to 119889119909 = 002 (119873 = 50) and vary the sizeof time step to plot the 119871

2norms of the relative errors in all

variables Figure 5(a) presents the rate of convergence versus

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

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Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

2 Journal of Computational Engineering

for high order derivatives in time and space in solving a 3Dheat transport problem at the microscale Sun and Zhang[12] proposed a new high order finite difference discretizaionstrategy based on the Richardson extrapolation techniqueand on an operator interpolation scheme for solving one- andtwo-dimensional convection-diffusion equations

Compactmethods arewidely used in applications in engi-neering and physics Ekaterinaris [13] presents fourth orderfactorized upwind compact schemes for time accurate solu-tions of hyperbolic equation systems Li et al [14] obtainedsolutions of Navier-Stokes equation expressed in the streamfunction-vorticity form with a fourth order compact schemefor a driven cavity problem Shang [15] solved the time-dependent Maxwell equation with spurious high-frequencyoscillatory components of the numerical solution beingsuppressed by a spatial filter Spotz and Carey extended high-order compact schemes to time-dependent problems [16]El-Zoheiry [17] devised a three-level iterative scheme basedon the compact implicit method for solving the improvedBoussinesq equation Zhang et al [18] derived a fourthorder compact scheme with multigrid mesh refinement forconvection diffusion problems Central and upwind compactmethods were analyzed by Sengupta et al [19] and schemeswith improved boundary closure and an explicit high-orderupwind schemewere proposed Shen et al [20] demonstrateda new way of constructing high accuracy shock-capturingcompact schemes Jocksch et al [21] developed a fifth ordercompact upwind-biased finite difference scheme for com-pressible Couette flow Mohamad et al [22] presented a thirdorder compact upwind scheme for flows containing disconti-nuities Anthonio and Hall [23] simulated free surface flowswith a high-order compact finite difference method and afourth order Runge-Kutta method for temporal discretiza-tion Cruz and Chen [24] modeled porous beds with highorder time-marching fourth order finite difference schemesA dual-compact scheme with the increased dispersive accu-racy for the convective terms was proposed byChiu and Sheu[25] A new family of high-order compact upwind differenceschemes with good spectral resolution was introduced byZhou et al [26] Sanyasiraju andMishra [27] developed sixthorder compact schemes for convection diffusion equationswith constant coefficients Dehghan and Mohebbi [28] pre-sented unconditionally stable fourth order accuracy in bothspace and time solution to unsteady convection diffusionequation Shah et al [29] solved the Navier-Stokes equationsin which the convective terms were approximated with athird order upwind compact scheme using flux differencingsplitting Xie et al [30] presented the numerical solution ofthe one dimensional nonlinear Schrodinger equation Tonelliand Petti [31] used a finite volume technique to solve theadvective part of the Boussinesq equations and applied a finitedifference method to discretize dispersive and source termsThe time integration was performed with the fourth orderAdams-Bashforth-Moulton method

Irregular geometries could hinder the application ofcompact methods To facilitate nonuniform discretizationone way is to use a transformation so that compact methodscan be implemented in the transformed space where thenominal rate of accuracy could be achieved For example

zeros of Chebyshev polynomials could be adopted as the gridpoints for computation [32 33] Gamet et al [34] developeda 4th order compact scheme for approximating the firstorder derivative with a full inclusion of metrics directly ona nonuniform mesh with variable grid spacing Using thismethod a direct numerical simulation of a compressible tur-bulent channel flow was conducted Vasilyev [35] presentedhigh order finite difference simulation of turbulent flowson nonuniform meshes with good conservation propertiesHermanns and Hernandez [36] developed high order finitedifference methods on nonuniform grids Usually the actualaccuracy is determined by the effect of the transformationAnother option for nonuniform grids is to use full inclusionof metrics (FIM) such as Liu et al [33] where fourth andsixth order compact methods were used for parabolic partialdifferential equations encountered in geophysical modelingGenerally speaking FIM could be difficult for deriving highaccuracy schemes

In the past two decades Boussinesq-type equationsderived fromdepth-integration of conservation laws formassandmomentum have beenwidely used to simulate nonlinearwater waves in coastal and ocean engineering One of thechallenges of solving this class of partial differential equationsis the numerical treatment of the dispersive terms describedby the mixed third order derivative in space (twice) and time

One objective of this paper is to develop 6th ordercoupled compact schemes formodified Boussinesq equationswith analytical exact solution for comparisonThese schemesdemonstrate high order accuracy and fast convergence ratesin both space and time The idea in this paper could be usedfor solving other systems of coupled nonlinear equations

This paper is outlined as below In simulation approachesthe modified Boussinesq equations and their exact solutionscorresponding to specially devised forcing functions aregiven Then the solution ideas and procedures are presentedfollowed by schemes of sixth order accuracy developed forthese equations In simulation results the solutions of all cou-pled variables and the convergence rates in space and time aredemonstrated In addition the effect of long time integrationis investigated to show numerical stability over time Finallyconclusion summarises the novelty and uniqueness of theschemes developed in this paper and future work is planned

2 Simulation Approaches

21 Modified Boussinesq Equations In terms of the velocity119906 and water depth 120578 Boussinesq-type equations in onehorizontal dimension could bemodified into below equationswhich are similar to [37]

120597120578

120597119905

+

120597 (119906120578)

120597119909

= 119891 (119906 120578 119909 119905)

120597119906

120597119905

+ 119906

120597119906

120597119909

minus

120583

3

1205973119906

1205971199092120597119905

+

120597120578

120597119909

= 119892 (119906 120578 119909 119905)

(1)

where 120583 is a dimensionless parameter measuring the wavedispersion at a given still-water depth The above equationsare closed with two initial conditions one for 119906 and one for120578 and boundary conditions two for 119906 and one for 120578 This is

Journal of Computational Engineering 3

a system of nonlinearly coupled partial differential equationswith strong convection and dispersivity

To solve this system of equations by the method of lines[38] (1) could be formulated into a system of time-dependentequations in terms of 120578 and a new variable 119908

120597120578

120597119905

= 119865

120597119908

120597119905

= 119866

(2)

where

119865 = 119891 (119906 120578 119909 119905) minus

120597 (119906120578)

120597119909

= 119891 (119906 120578 119909 119905) minus 120578

120597119906

120597119909

minus 119906

120597120578

120597119909

(3)

119908 = 119906 minus

120583

3

1205972119906

1205971199092 119866 = 119892 (119906 120578 119909 119905) minus 119906

120597119906

120597119909

minus

120597120578

120597119909

(4)

Nowwe could take care of the systemof equations in time andspace separately To march (2) in time a variety of methodsare available [32 38 39] Since implicit methods are used inspace and the system is strongly nonlinear we select explicitmethods for time evolution In this paper up to third orderAdams-Bashforth methods are used to integrate in timeDuring the initial time steps lower order Adams-Bashforthmethods are used instead

In order to compare results and investigate the rate ofconvergence exact solutions to (1) are needed With somespecific forms of 119891 and 119892 in these equations with 120583 = 3 theexact solutions to (1) are available In this paper we presentthe special forms of 119891 and 119892 and the corresponding exactsolutions as given below

119891 = (2119909 + 1) 1198902119909minus2119905

minus 119890119909minus119905

119892 = (1199092+ 119909) 119890

2119909minus2119905+ 3119890119909minus119905

(5)

subject to the initial conditions

120578 (119909 0) = 119890119909 119906 (119909 0) = 119909119890

119909 (6)

and the boundary conditions

120578 (1 119905) = 1198901minus119905 119906 (1 119905) = 119890

1minus119905

120597119906 (2 119905)

120597119909

= 31198902minus119905

(7)

the exact solutions to (1) are

120578 = 119890119909minus119905

119906 = 119909119890119909minus119905

1 le 119909 le 2 119905 ge 0 (8)

22 Solution Ideas and Procedures Themajor steps in findingnumerical solution are elaborated below These steps involveCompact Schemes I II and III which are described in the nextsubsection We use superscripts to denote the time levels ofvariables

Based on the initial 119906(0) and 120578(0) we use Compact SchemeI to compute 120597119906

(0)120597119909 and 120597120578

(0)120597119909 then by (3) and (4)

to acquire 119865(0) and 119866(0) and we use Compact Scheme II to

compute 1205972119906(0)1205971199092 from 119906

(0) then by (4) to obtain 119908(0)

With 119865(0) 119866(0) 120578(0) and 119908(0) being known by the first orderAdams-Bashforth method 120578(1) and 119908(1) could be computedNext we use Compact Scheme III to compute 119906

(1) Thisconcludes the computation in time level one

Based on 119906(1) and 120578

(1) we use Compact Scheme Ito compute 120597119906(1)120597119909 and 120597120578

(1)120597119909 then by (3) and (4) to

acquire 119865(1) and 119866(1) With 119865(1) 119866(1) 119865(0) 119866(0) 120578(1) and 119908(1)being known by the second order Adams-Bashforthmethod120578(2) and 119908

(2) could be computed Next we use CompactScheme III to compute 119906(2) This finishes the computation intime level two

Based on 119906(2) and 120578

(2) repeat the same procedures anduse the third order Adams-Bashforthmethod to compute 120578(3)

and 119906(3) Continue the computations and use the third orderAdams-Bashforth method from now on to compute 119906 and 120578at the next time level

23 Coupled Compact Schemes of High Accuracy Now wefocus on the spatial discretization and the specific high ordercompact schemesmentioned earlierWepartition the domainof interest 1 le 119909 le 2 into119873 equal intervals with119873+1 nodeswith the nodal index starting from 0 to the last one119873

231 Compact Scheme I Solving for 120597119906120597119909120597120578120597119909withKnown119906 120578 Denote either 119906 or 120578 with 119884 The first derivative of 119884relative to 119909 at node 119909

119894is 1198841015840119894= 120597119884119894120597119909 The uniform grid

size is ℎ Based on the known values of 119906 and 120578 at the currenttime level (eg the initial values) 120597119906120597119909 and 120597120578120597119909 could becomputed accurately with compact methods For the interiornodes the scheme of the sixth order accuracy is

121198841015840

119894minus1+ 36119884

1015840

119894+ 12119884

1015840

119894+1

= ℎminus1(minus119884119894minus2

minus 28119884119894minus1

+ 28119884119894+1

+ 119884119894+2) + 119900 (ℎ

6)

(9)

For the two left end nodes use the sixth order scheme with119894 = 1 2 respectively

minus601198841015840

119894minus1minus 300119884

1015840

119894= ℎminus1(197119884

119894minus1+ 25119884

119894minus 300119884

119894+1

+100119884119894+2

minus 25119884119894+3

+ 3119884119894+4)

+ 119900 (ℎ6)

(10)

For the two right end nodes when solving for 120597120578120597119909 use thesixth order scheme below with 119894 = 119873119873 minus 1 respectivelywhen solving for 120597119906120597119909 use the same scheme with 119894 = 119873 minus

1 and the Neumann boundary condition

minus 601198841015840

119894minus 300119884

1015840

119894+1

= ℎminus1(minus197119884

119894minus 25119884

119894minus1+ 300119884

119894minus2minus 100119884

119894minus3

+25119884119894minus4

minus 3119884119894minus5) + 119900 (ℎ

6)

(11)

Equations (9) (10) and (11) form a tridiagonal system ofdimension119873+ 1 by119873+ 1 in terms of the nodal values of the

4 Journal of Computational Engineering

first derivative 1198841015840 The coefficient matrix of the linear systemis stored as three vectors in order to save storage The linearsystem is solved with theThomas Algorithm [40]

232 Compact Scheme II Solving for 12059721199061205971199092 with Known 119906Denote 119906 with 119884 11988410158401015840

119894= 12059721198841198941205971199092 Based on the known values

of 119906 at the current time level (eg the initial time) 12059721199061205971199092could be computed with high order compact methods Forthe interior nodes the scheme of the sixth order accuracy is

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1

= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(12)

For the two left end nodes use the following sixth orderscheme with 119894 = 1 2 respectively with the Dirichletboundary condition 119884

0= 1198901minus119905 at 119909 = 1 included

12057211988410158401015840

119894minus1+ 12057311988410158401015840

119894= ℎminus2(119887119884119894minus1

+ 1198860119884119894+ 1198861119884119894+1

+ 1198862119884119894+2

+ 1198863119884119894+3

+ 1198864119884119894+4

+1198865119884119894+5) + 119900 (ℎ

6)

(13)

where 120572 = 2823(13) 120573 = 32340 119887 = 37350(1927) 1198860=

minus75537 1198861= 36767(12) 119886

2= 4762(1627) 119886

3= minus4620

1198864= 1463 and 119886

5= (1154) minus 187 For the right end node

use the fifth order mixed scheme as below with the Neumannboundary condition 1198841015840

119873= 31198902minus119905 at 119909 = 2 included

7211988410158401015840

119873minus1+ 18119884

10158401015840

119873minus2= minus 30ℎ

minus11198841015840

119873

+ ℎminus2(127119884

119873minus 216119884

119873minus1

+81119884119873minus2

+ 8119884119873minus3

) + 119900 (ℎ5)

(14)

For the node next to the right end node use the sixth orderscheme below with 119894 = 119873 and the same coefficients as in (13)

12057311988410158401015840

119894minus1+ 12057211988410158401015840

119894= ℎminus2(1198865119884119894minus6

+ 1198864119884119894minus5

+ 1198863119884119894minus4

+ 1198862119884119894minus3

+ 1198861119884119894minus2

+ 1198860119884119894minus1

+119887119884119894) + 119900 (ℎ

6)

(15)

Equations (12) (13) (14) and (15) form a quasitridiagonalsystem of dimension 119873 + 1 by 119873 + 1 in terms of the nodalvalues of the second derivative of 119906 since only one nonzeroelement in the last row lies outside the tridiagonal lines Thecoefficient matrix of the linear system could still be storedas three vectors The linear system is solved by the ThomasAlgorithm with a minor modification

233 Compact Scheme III Solving for 119906 12059721199061205971199092 with Known119908 After the time integration with the Adams-Bashforthmethod 119908 at the next time level becomes known Now we

compute the new nodal values of 119906 and 12059721199061205971199092 at the nexttime level with 119908 based on (4)

119906 minus

1205972119906

1205971199092= 119908 (16)

Since two variables are solved for at the same time anotherequation is needed here A compact scheme of the sixth orderaccuracy as below could be the supplementary equation

sum

119894

119886119894

1205972119906119894

1205971199092=

1

ℎ2sum

119896

119887119896119906119896+ 119900 (ℎ

6) (17)

Therefore the resulting system is of dimension 2119873 + 2 by2119873+2 in terms of the nodal values of 12059721199061205971199092 and 119906 In otherwords there are two equations for each node Denote 119906 with119884 at the node 119894 119906

119894= 119884119894 12059721199061198941205971199092= 12059721198841198941205971199092= 11988410158401015840

119894 For the

interior nodes we choose (17) to be (12) Then the system of(16) and (17) at the interior nodes becomes

minus11988410158401015840

119894+ 119884119894= 119908119894 (18)

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(19)

For the node next to the very end node on each side (18) with119894 = 1 for the left and 119894 = 119873minus1 for the right is implementedThesecond equation at the same node is the sixth order schemeas below with 119894 = 3 for the left one and 119894 = 119873minus1 for the rightone

11988410158401015840

119894minus3minus 24119884

10158401015840

119894minus2minus 194119884

10158401015840

119894minus1minus 24119884

10158401015840

119894+ 11988410158401015840

119894+1

= ℎminus2(minus240119884

119894minus2+ 480119884

119894minus1minus 240119884

119894) + 119900 (ℎ

6)

(20)

For the left end node at 119909 = 1 the Dirichlet boundarycondition for 119884 is implemented The second equation at thesame node is (13) with 119894 = 1 and the Dirichlet boundarycondition appended For the right end node at 119909 = 2 (18)is implemented with 119894 = 119873 The second equation at the samenode is the fifth order mixed scheme that is (14) with 119894 = 119873

and the Neumann boundary condition1198841015840119873= 31198902minus119905 appended

We specially arranged 11988410158401015840

119894and 119884

119894alternately that is 11988410158401015840

0

11988401198841015840101584011198841 to acquire a banded linear system of dimension

2119873 + 2 by 2119873 + 2 in terms of the nodal values of 12059721199061205971199092and 119906 Without such deliberate alternating arrangement thecoefficientmatrixwould be blockedwith sparsityThis systemwith a small bandwidth could be efficiently solved via thebanded Gaussian elimination with scaled partial pivoting

3 Simulation Results

31 Approximate and Exact Solutions We compare ourapproximate (numerical) solutions with exact solutionsfor 119906 and its derivatives in Figure 1(a) 120578 and its derivativein Figure 1(b) at the dimensionless time 119879 = 01 The timestep size is 119889119905 = 00001 and the space step size is 119889119909 = 005No visual difference could be seen from these figures betweenthe approximate and exact solutions To investigate errors weneed to check convergence rates in both space and time

Journal of Computational Engineering 5

x

Valu

e

1 125 15 175 2

4

8

12

16

20

24

28

ApproxuApproxuxApprox uxx

Exact uExact uxExact uxx

(a)

x

1 125 15 175 2

+ + + + ++

++

++

++

++

++

++

++

+

Valu

e

3

4

5

6

7

+Approx120578Exact 120578

Approx120578xExact 120578x

(b)

Figure 1 Comparison of the approximate and exact solutions for 119906 (a) and 120578 (b)

50 100 150 200

Conv rate

10minus13

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

N

ux = 623

120578x

ux

120578x

= 612

(a)

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

L2

(rel

ativ

e err

or)

50 100 150 200N

Conv rate uxx = 593

(b)

Figure 2 Convergence rates for Compact Scheme I (a) and Compact Scheme II (b) in space

32 Convergence Rates of Separate Compact Schemes in Space

321 Compact Scheme I The performance of CompactScheme I is examined here Since the exact solutions areavailable we compute the Euclidean (119871

2) norms of the

relative errors for the first derivative of 119906 and 120578 at all nodesFigure 2(a) presents in the log-log scale the 119871

2norm of

the relative errors versus the number of nodes used indiscretizationWhen the spatial resolution goes down to 001that is 119873 = 100 the Euclidean norms of the relative errors

6 Journal of Computational Engineering

50 100 150 200N

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4L2

(rel

ativ

e err

or)

Conv rateu = 557

uxx = 610

(a)

50 100 150 200N

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

Conv rateu = 630

ux = 581

uxx = 484

120578 = 682

120578x = 537

dt = 10minus4

(b)

Figure 3 (a) Convergence rates for Compact Scheme III (b) Convergence rates for combined schemes in space with 119889119905 = 10minus4

drop to themachine zero in double precisionThe data pointsdemonstrate the sixth order convergence rate for both 120597119906120597119909and 120597120578119889119909

322 Compact Scheme II To see the performance of Com-pact Scheme II we use the known exact solution to computethe Euclidean norm of the relative errors for 12059721199061205971199092 at allnodes is In the log-log scale Figure 2(b) shows the 119871

2norm

of the relative error versus the number of nodes used in spaceBefore the spatial step reduces to 001 the 119871

2norm of the

relative errors of 12059721199061205971199092 drops to the machine zero Thisfigure clearly demonstrates the sixth order convergence rate

323 Compact Scheme III The performance of CompactScheme III is demonstrated below Taking advantage of theknown exact solution we compute the Euclidean normsof the relative errors for both 119906 and 120597

21199061205971199092 at all nodes

Figure 3(a) plots the 1198712norms of the relative errors versus

the number of nodes used in discretization The linearsystem generated from the Compact Scheme III is no longertridiagonal but it is still banded Because a fifth order schemeis used at the right end node 119894 = 119873 in order to adopt theNeumann boundary condition the convergence rates for 119906and 12059721199061205971199092 are slightly different but about sixth order

33 Convergence Rates of Combined Schemes in Space

331 Time Step 119889119905 = 10minus4 Nowwe examine the convergence

rates for all three compact schemes together over 1000 timesteps after the initial conditions are imposed During thisperiod of time Compact Schemes I II and III are used one

after another to solve these modified Boussinesq equationsFigure 3(b) presents the overall spatial convergence rates(indicated in the legend resp) for all variables after 1000 timesteps which was set to be 119889119905 = 10

minus4 The lowest convergencerate of 484 is observed for 12059721199061205971199092 because the variable 119906carries a second derivative and the boundary condition is notexplicitly given in terms of 12059721199061205971199092 The rest of the variableshave the sixth order convergence rate

332 Time Step 119889119905 = 10minus5 and 119889119905 = 10

minus6 Similar to Figures3(b) and 4(a) with 119889119905 = 10

minus5 and Figure 4(b) with 119889119905 = 10minus6

present the spatial convergence rates (indicated in the legend)involving Compact Schemes I II and III for all variables after1000 time steps The lowest convergence rate about 49 isexpected for 12059721199061205971199092 for the same reason as before For therest variables the sixth order convergence rates are clearlydemonstrated From Figures 3(b) 4(a) and 4(b) we observethat as the time step size reduces high order convergencerates are maintained for higher spatial resolution (larger119873 infigures)This is because a smaller time step size produces lesserror during time integration However as the spatial resolu-tion (119873) increases the rounding error increases accordinglyand eventually overtakes the truncation error Therefore wedo not see the errors keep decreasing as119873 increases

34 Convergence Rate in Time Integration After we exam-ined the spatial convergence rates now we demonstrate theconvergence rate in time integration To this end we fix thespatial step size to 119889119909 = 002 (119873 = 50) and vary the sizeof time step to plot the 119871

2norms of the relative errors in all

variables Figure 5(a) presents the rate of convergence versus

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

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International Journal of

Journal of Computational Engineering 3

a system of nonlinearly coupled partial differential equationswith strong convection and dispersivity

To solve this system of equations by the method of lines[38] (1) could be formulated into a system of time-dependentequations in terms of 120578 and a new variable 119908

120597120578

120597119905

= 119865

120597119908

120597119905

= 119866

(2)

where

119865 = 119891 (119906 120578 119909 119905) minus

120597 (119906120578)

120597119909

= 119891 (119906 120578 119909 119905) minus 120578

120597119906

120597119909

minus 119906

120597120578

120597119909

(3)

119908 = 119906 minus

120583

3

1205972119906

1205971199092 119866 = 119892 (119906 120578 119909 119905) minus 119906

120597119906

120597119909

minus

120597120578

120597119909

(4)

Nowwe could take care of the systemof equations in time andspace separately To march (2) in time a variety of methodsare available [32 38 39] Since implicit methods are used inspace and the system is strongly nonlinear we select explicitmethods for time evolution In this paper up to third orderAdams-Bashforth methods are used to integrate in timeDuring the initial time steps lower order Adams-Bashforthmethods are used instead

In order to compare results and investigate the rate ofconvergence exact solutions to (1) are needed With somespecific forms of 119891 and 119892 in these equations with 120583 = 3 theexact solutions to (1) are available In this paper we presentthe special forms of 119891 and 119892 and the corresponding exactsolutions as given below

119891 = (2119909 + 1) 1198902119909minus2119905

minus 119890119909minus119905

119892 = (1199092+ 119909) 119890

2119909minus2119905+ 3119890119909minus119905

(5)

subject to the initial conditions

120578 (119909 0) = 119890119909 119906 (119909 0) = 119909119890

119909 (6)

and the boundary conditions

120578 (1 119905) = 1198901minus119905 119906 (1 119905) = 119890

1minus119905

120597119906 (2 119905)

120597119909

= 31198902minus119905

(7)

the exact solutions to (1) are

120578 = 119890119909minus119905

119906 = 119909119890119909minus119905

1 le 119909 le 2 119905 ge 0 (8)

22 Solution Ideas and Procedures Themajor steps in findingnumerical solution are elaborated below These steps involveCompact Schemes I II and III which are described in the nextsubsection We use superscripts to denote the time levels ofvariables

Based on the initial 119906(0) and 120578(0) we use Compact SchemeI to compute 120597119906

(0)120597119909 and 120597120578

(0)120597119909 then by (3) and (4)

to acquire 119865(0) and 119866(0) and we use Compact Scheme II to

compute 1205972119906(0)1205971199092 from 119906

(0) then by (4) to obtain 119908(0)

With 119865(0) 119866(0) 120578(0) and 119908(0) being known by the first orderAdams-Bashforth method 120578(1) and 119908(1) could be computedNext we use Compact Scheme III to compute 119906

(1) Thisconcludes the computation in time level one

Based on 119906(1) and 120578

(1) we use Compact Scheme Ito compute 120597119906(1)120597119909 and 120597120578

(1)120597119909 then by (3) and (4) to

acquire 119865(1) and 119866(1) With 119865(1) 119866(1) 119865(0) 119866(0) 120578(1) and 119908(1)being known by the second order Adams-Bashforthmethod120578(2) and 119908

(2) could be computed Next we use CompactScheme III to compute 119906(2) This finishes the computation intime level two

Based on 119906(2) and 120578

(2) repeat the same procedures anduse the third order Adams-Bashforthmethod to compute 120578(3)

and 119906(3) Continue the computations and use the third orderAdams-Bashforth method from now on to compute 119906 and 120578at the next time level

23 Coupled Compact Schemes of High Accuracy Now wefocus on the spatial discretization and the specific high ordercompact schemesmentioned earlierWepartition the domainof interest 1 le 119909 le 2 into119873 equal intervals with119873+1 nodeswith the nodal index starting from 0 to the last one119873

231 Compact Scheme I Solving for 120597119906120597119909120597120578120597119909withKnown119906 120578 Denote either 119906 or 120578 with 119884 The first derivative of 119884relative to 119909 at node 119909

119894is 1198841015840119894= 120597119884119894120597119909 The uniform grid

size is ℎ Based on the known values of 119906 and 120578 at the currenttime level (eg the initial values) 120597119906120597119909 and 120597120578120597119909 could becomputed accurately with compact methods For the interiornodes the scheme of the sixth order accuracy is

121198841015840

119894minus1+ 36119884

1015840

119894+ 12119884

1015840

119894+1

= ℎminus1(minus119884119894minus2

minus 28119884119894minus1

+ 28119884119894+1

+ 119884119894+2) + 119900 (ℎ

6)

(9)

For the two left end nodes use the sixth order scheme with119894 = 1 2 respectively

minus601198841015840

119894minus1minus 300119884

1015840

119894= ℎminus1(197119884

119894minus1+ 25119884

119894minus 300119884

119894+1

+100119884119894+2

minus 25119884119894+3

+ 3119884119894+4)

+ 119900 (ℎ6)

(10)

For the two right end nodes when solving for 120597120578120597119909 use thesixth order scheme below with 119894 = 119873119873 minus 1 respectivelywhen solving for 120597119906120597119909 use the same scheme with 119894 = 119873 minus

1 and the Neumann boundary condition

minus 601198841015840

119894minus 300119884

1015840

119894+1

= ℎminus1(minus197119884

119894minus 25119884

119894minus1+ 300119884

119894minus2minus 100119884

119894minus3

+25119884119894minus4

minus 3119884119894minus5) + 119900 (ℎ

6)

(11)

Equations (9) (10) and (11) form a tridiagonal system ofdimension119873+ 1 by119873+ 1 in terms of the nodal values of the

4 Journal of Computational Engineering

first derivative 1198841015840 The coefficient matrix of the linear systemis stored as three vectors in order to save storage The linearsystem is solved with theThomas Algorithm [40]

232 Compact Scheme II Solving for 12059721199061205971199092 with Known 119906Denote 119906 with 119884 11988410158401015840

119894= 12059721198841198941205971199092 Based on the known values

of 119906 at the current time level (eg the initial time) 12059721199061205971199092could be computed with high order compact methods Forthe interior nodes the scheme of the sixth order accuracy is

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1

= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(12)

For the two left end nodes use the following sixth orderscheme with 119894 = 1 2 respectively with the Dirichletboundary condition 119884

0= 1198901minus119905 at 119909 = 1 included

12057211988410158401015840

119894minus1+ 12057311988410158401015840

119894= ℎminus2(119887119884119894minus1

+ 1198860119884119894+ 1198861119884119894+1

+ 1198862119884119894+2

+ 1198863119884119894+3

+ 1198864119884119894+4

+1198865119884119894+5) + 119900 (ℎ

6)

(13)

where 120572 = 2823(13) 120573 = 32340 119887 = 37350(1927) 1198860=

minus75537 1198861= 36767(12) 119886

2= 4762(1627) 119886

3= minus4620

1198864= 1463 and 119886

5= (1154) minus 187 For the right end node

use the fifth order mixed scheme as below with the Neumannboundary condition 1198841015840

119873= 31198902minus119905 at 119909 = 2 included

7211988410158401015840

119873minus1+ 18119884

10158401015840

119873minus2= minus 30ℎ

minus11198841015840

119873

+ ℎminus2(127119884

119873minus 216119884

119873minus1

+81119884119873minus2

+ 8119884119873minus3

) + 119900 (ℎ5)

(14)

For the node next to the right end node use the sixth orderscheme below with 119894 = 119873 and the same coefficients as in (13)

12057311988410158401015840

119894minus1+ 12057211988410158401015840

119894= ℎminus2(1198865119884119894minus6

+ 1198864119884119894minus5

+ 1198863119884119894minus4

+ 1198862119884119894minus3

+ 1198861119884119894minus2

+ 1198860119884119894minus1

+119887119884119894) + 119900 (ℎ

6)

(15)

Equations (12) (13) (14) and (15) form a quasitridiagonalsystem of dimension 119873 + 1 by 119873 + 1 in terms of the nodalvalues of the second derivative of 119906 since only one nonzeroelement in the last row lies outside the tridiagonal lines Thecoefficient matrix of the linear system could still be storedas three vectors The linear system is solved by the ThomasAlgorithm with a minor modification

233 Compact Scheme III Solving for 119906 12059721199061205971199092 with Known119908 After the time integration with the Adams-Bashforthmethod 119908 at the next time level becomes known Now we

compute the new nodal values of 119906 and 12059721199061205971199092 at the nexttime level with 119908 based on (4)

119906 minus

1205972119906

1205971199092= 119908 (16)

Since two variables are solved for at the same time anotherequation is needed here A compact scheme of the sixth orderaccuracy as below could be the supplementary equation

sum

119894

119886119894

1205972119906119894

1205971199092=

1

ℎ2sum

119896

119887119896119906119896+ 119900 (ℎ

6) (17)

Therefore the resulting system is of dimension 2119873 + 2 by2119873+2 in terms of the nodal values of 12059721199061205971199092 and 119906 In otherwords there are two equations for each node Denote 119906 with119884 at the node 119894 119906

119894= 119884119894 12059721199061198941205971199092= 12059721198841198941205971199092= 11988410158401015840

119894 For the

interior nodes we choose (17) to be (12) Then the system of(16) and (17) at the interior nodes becomes

minus11988410158401015840

119894+ 119884119894= 119908119894 (18)

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(19)

For the node next to the very end node on each side (18) with119894 = 1 for the left and 119894 = 119873minus1 for the right is implementedThesecond equation at the same node is the sixth order schemeas below with 119894 = 3 for the left one and 119894 = 119873minus1 for the rightone

11988410158401015840

119894minus3minus 24119884

10158401015840

119894minus2minus 194119884

10158401015840

119894minus1minus 24119884

10158401015840

119894+ 11988410158401015840

119894+1

= ℎminus2(minus240119884

119894minus2+ 480119884

119894minus1minus 240119884

119894) + 119900 (ℎ

6)

(20)

For the left end node at 119909 = 1 the Dirichlet boundarycondition for 119884 is implemented The second equation at thesame node is (13) with 119894 = 1 and the Dirichlet boundarycondition appended For the right end node at 119909 = 2 (18)is implemented with 119894 = 119873 The second equation at the samenode is the fifth order mixed scheme that is (14) with 119894 = 119873

and the Neumann boundary condition1198841015840119873= 31198902minus119905 appended

We specially arranged 11988410158401015840

119894and 119884

119894alternately that is 11988410158401015840

0

11988401198841015840101584011198841 to acquire a banded linear system of dimension

2119873 + 2 by 2119873 + 2 in terms of the nodal values of 12059721199061205971199092and 119906 Without such deliberate alternating arrangement thecoefficientmatrixwould be blockedwith sparsityThis systemwith a small bandwidth could be efficiently solved via thebanded Gaussian elimination with scaled partial pivoting

3 Simulation Results

31 Approximate and Exact Solutions We compare ourapproximate (numerical) solutions with exact solutionsfor 119906 and its derivatives in Figure 1(a) 120578 and its derivativein Figure 1(b) at the dimensionless time 119879 = 01 The timestep size is 119889119905 = 00001 and the space step size is 119889119909 = 005No visual difference could be seen from these figures betweenthe approximate and exact solutions To investigate errors weneed to check convergence rates in both space and time

Journal of Computational Engineering 5

x

Valu

e

1 125 15 175 2

4

8

12

16

20

24

28

ApproxuApproxuxApprox uxx

Exact uExact uxExact uxx

(a)

x

1 125 15 175 2

+ + + + ++

++

++

++

++

++

++

++

+

Valu

e

3

4

5

6

7

+Approx120578Exact 120578

Approx120578xExact 120578x

(b)

Figure 1 Comparison of the approximate and exact solutions for 119906 (a) and 120578 (b)

50 100 150 200

Conv rate

10minus13

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

N

ux = 623

120578x

ux

120578x

= 612

(a)

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

L2

(rel

ativ

e err

or)

50 100 150 200N

Conv rate uxx = 593

(b)

Figure 2 Convergence rates for Compact Scheme I (a) and Compact Scheme II (b) in space

32 Convergence Rates of Separate Compact Schemes in Space

321 Compact Scheme I The performance of CompactScheme I is examined here Since the exact solutions areavailable we compute the Euclidean (119871

2) norms of the

relative errors for the first derivative of 119906 and 120578 at all nodesFigure 2(a) presents in the log-log scale the 119871

2norm of

the relative errors versus the number of nodes used indiscretizationWhen the spatial resolution goes down to 001that is 119873 = 100 the Euclidean norms of the relative errors

6 Journal of Computational Engineering

50 100 150 200N

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4L2

(rel

ativ

e err

or)

Conv rateu = 557

uxx = 610

(a)

50 100 150 200N

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

Conv rateu = 630

ux = 581

uxx = 484

120578 = 682

120578x = 537

dt = 10minus4

(b)

Figure 3 (a) Convergence rates for Compact Scheme III (b) Convergence rates for combined schemes in space with 119889119905 = 10minus4

drop to themachine zero in double precisionThe data pointsdemonstrate the sixth order convergence rate for both 120597119906120597119909and 120597120578119889119909

322 Compact Scheme II To see the performance of Com-pact Scheme II we use the known exact solution to computethe Euclidean norm of the relative errors for 12059721199061205971199092 at allnodes is In the log-log scale Figure 2(b) shows the 119871

2norm

of the relative error versus the number of nodes used in spaceBefore the spatial step reduces to 001 the 119871

2norm of the

relative errors of 12059721199061205971199092 drops to the machine zero Thisfigure clearly demonstrates the sixth order convergence rate

323 Compact Scheme III The performance of CompactScheme III is demonstrated below Taking advantage of theknown exact solution we compute the Euclidean normsof the relative errors for both 119906 and 120597

21199061205971199092 at all nodes

Figure 3(a) plots the 1198712norms of the relative errors versus

the number of nodes used in discretization The linearsystem generated from the Compact Scheme III is no longertridiagonal but it is still banded Because a fifth order schemeis used at the right end node 119894 = 119873 in order to adopt theNeumann boundary condition the convergence rates for 119906and 12059721199061205971199092 are slightly different but about sixth order

33 Convergence Rates of Combined Schemes in Space

331 Time Step 119889119905 = 10minus4 Nowwe examine the convergence

rates for all three compact schemes together over 1000 timesteps after the initial conditions are imposed During thisperiod of time Compact Schemes I II and III are used one

after another to solve these modified Boussinesq equationsFigure 3(b) presents the overall spatial convergence rates(indicated in the legend resp) for all variables after 1000 timesteps which was set to be 119889119905 = 10

minus4 The lowest convergencerate of 484 is observed for 12059721199061205971199092 because the variable 119906carries a second derivative and the boundary condition is notexplicitly given in terms of 12059721199061205971199092 The rest of the variableshave the sixth order convergence rate

332 Time Step 119889119905 = 10minus5 and 119889119905 = 10

minus6 Similar to Figures3(b) and 4(a) with 119889119905 = 10

minus5 and Figure 4(b) with 119889119905 = 10minus6

present the spatial convergence rates (indicated in the legend)involving Compact Schemes I II and III for all variables after1000 time steps The lowest convergence rate about 49 isexpected for 12059721199061205971199092 for the same reason as before For therest variables the sixth order convergence rates are clearlydemonstrated From Figures 3(b) 4(a) and 4(b) we observethat as the time step size reduces high order convergencerates are maintained for higher spatial resolution (larger119873 infigures)This is because a smaller time step size produces lesserror during time integration However as the spatial resolu-tion (119873) increases the rounding error increases accordinglyand eventually overtakes the truncation error Therefore wedo not see the errors keep decreasing as119873 increases

34 Convergence Rate in Time Integration After we exam-ined the spatial convergence rates now we demonstrate theconvergence rate in time integration To this end we fix thespatial step size to 119889119909 = 002 (119873 = 50) and vary the sizeof time step to plot the 119871

2norms of the relative errors in all

variables Figure 5(a) presents the rate of convergence versus

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

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International Journal of

4 Journal of Computational Engineering

first derivative 1198841015840 The coefficient matrix of the linear systemis stored as three vectors in order to save storage The linearsystem is solved with theThomas Algorithm [40]

232 Compact Scheme II Solving for 12059721199061205971199092 with Known 119906Denote 119906 with 119884 11988410158401015840

119894= 12059721198841198941205971199092 Based on the known values

of 119906 at the current time level (eg the initial time) 12059721199061205971199092could be computed with high order compact methods Forthe interior nodes the scheme of the sixth order accuracy is

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1

= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(12)

For the two left end nodes use the following sixth orderscheme with 119894 = 1 2 respectively with the Dirichletboundary condition 119884

0= 1198901minus119905 at 119909 = 1 included

12057211988410158401015840

119894minus1+ 12057311988410158401015840

119894= ℎminus2(119887119884119894minus1

+ 1198860119884119894+ 1198861119884119894+1

+ 1198862119884119894+2

+ 1198863119884119894+3

+ 1198864119884119894+4

+1198865119884119894+5) + 119900 (ℎ

6)

(13)

where 120572 = 2823(13) 120573 = 32340 119887 = 37350(1927) 1198860=

minus75537 1198861= 36767(12) 119886

2= 4762(1627) 119886

3= minus4620

1198864= 1463 and 119886

5= (1154) minus 187 For the right end node

use the fifth order mixed scheme as below with the Neumannboundary condition 1198841015840

119873= 31198902minus119905 at 119909 = 2 included

7211988410158401015840

119873minus1+ 18119884

10158401015840

119873minus2= minus 30ℎ

minus11198841015840

119873

+ ℎminus2(127119884

119873minus 216119884

119873minus1

+81119884119873minus2

+ 8119884119873minus3

) + 119900 (ℎ5)

(14)

For the node next to the right end node use the sixth orderscheme below with 119894 = 119873 and the same coefficients as in (13)

12057311988410158401015840

119894minus1+ 12057211988410158401015840

119894= ℎminus2(1198865119884119894minus6

+ 1198864119884119894minus5

+ 1198863119884119894minus4

+ 1198862119884119894minus3

+ 1198861119884119894minus2

+ 1198860119884119894minus1

+119887119884119894) + 119900 (ℎ

6)

(15)

Equations (12) (13) (14) and (15) form a quasitridiagonalsystem of dimension 119873 + 1 by 119873 + 1 in terms of the nodalvalues of the second derivative of 119906 since only one nonzeroelement in the last row lies outside the tridiagonal lines Thecoefficient matrix of the linear system could still be storedas three vectors The linear system is solved by the ThomasAlgorithm with a minor modification

233 Compact Scheme III Solving for 119906 12059721199061205971199092 with Known119908 After the time integration with the Adams-Bashforthmethod 119908 at the next time level becomes known Now we

compute the new nodal values of 119906 and 12059721199061205971199092 at the nexttime level with 119908 based on (4)

119906 minus

1205972119906

1205971199092= 119908 (16)

Since two variables are solved for at the same time anotherequation is needed here A compact scheme of the sixth orderaccuracy as below could be the supplementary equation

sum

119894

119886119894

1205972119906119894

1205971199092=

1

ℎ2sum

119896

119887119896119906119896+ 119900 (ℎ

6) (17)

Therefore the resulting system is of dimension 2119873 + 2 by2119873+2 in terms of the nodal values of 12059721199061205971199092 and 119906 In otherwords there are two equations for each node Denote 119906 with119884 at the node 119894 119906

119894= 119884119894 12059721199061198941205971199092= 12059721198841198941205971199092= 11988410158401015840

119894 For the

interior nodes we choose (17) to be (12) Then the system of(16) and (17) at the interior nodes becomes

minus11988410158401015840

119894+ 119884119894= 119908119894 (18)

811988410158401015840

119894minus1+ 44119884

10158401015840

119894+ 811988410158401015840

119894+1= ℎminus2(3119884119894minus2

+ 48119884119894minus1

minus 102119884119894

+48119884119894+1

+ 3119884119894+2) + 119900 (ℎ

6)

(19)

For the node next to the very end node on each side (18) with119894 = 1 for the left and 119894 = 119873minus1 for the right is implementedThesecond equation at the same node is the sixth order schemeas below with 119894 = 3 for the left one and 119894 = 119873minus1 for the rightone

11988410158401015840

119894minus3minus 24119884

10158401015840

119894minus2minus 194119884

10158401015840

119894minus1minus 24119884

10158401015840

119894+ 11988410158401015840

119894+1

= ℎminus2(minus240119884

119894minus2+ 480119884

119894minus1minus 240119884

119894) + 119900 (ℎ

6)

(20)

For the left end node at 119909 = 1 the Dirichlet boundarycondition for 119884 is implemented The second equation at thesame node is (13) with 119894 = 1 and the Dirichlet boundarycondition appended For the right end node at 119909 = 2 (18)is implemented with 119894 = 119873 The second equation at the samenode is the fifth order mixed scheme that is (14) with 119894 = 119873

and the Neumann boundary condition1198841015840119873= 31198902minus119905 appended

We specially arranged 11988410158401015840

119894and 119884

119894alternately that is 11988410158401015840

0

11988401198841015840101584011198841 to acquire a banded linear system of dimension

2119873 + 2 by 2119873 + 2 in terms of the nodal values of 12059721199061205971199092and 119906 Without such deliberate alternating arrangement thecoefficientmatrixwould be blockedwith sparsityThis systemwith a small bandwidth could be efficiently solved via thebanded Gaussian elimination with scaled partial pivoting

3 Simulation Results

31 Approximate and Exact Solutions We compare ourapproximate (numerical) solutions with exact solutionsfor 119906 and its derivatives in Figure 1(a) 120578 and its derivativein Figure 1(b) at the dimensionless time 119879 = 01 The timestep size is 119889119905 = 00001 and the space step size is 119889119909 = 005No visual difference could be seen from these figures betweenthe approximate and exact solutions To investigate errors weneed to check convergence rates in both space and time

Journal of Computational Engineering 5

x

Valu

e

1 125 15 175 2

4

8

12

16

20

24

28

ApproxuApproxuxApprox uxx

Exact uExact uxExact uxx

(a)

x

1 125 15 175 2

+ + + + ++

++

++

++

++

++

++

++

+

Valu

e

3

4

5

6

7

+Approx120578Exact 120578

Approx120578xExact 120578x

(b)

Figure 1 Comparison of the approximate and exact solutions for 119906 (a) and 120578 (b)

50 100 150 200

Conv rate

10minus13

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

N

ux = 623

120578x

ux

120578x

= 612

(a)

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

L2

(rel

ativ

e err

or)

50 100 150 200N

Conv rate uxx = 593

(b)

Figure 2 Convergence rates for Compact Scheme I (a) and Compact Scheme II (b) in space

32 Convergence Rates of Separate Compact Schemes in Space

321 Compact Scheme I The performance of CompactScheme I is examined here Since the exact solutions areavailable we compute the Euclidean (119871

2) norms of the

relative errors for the first derivative of 119906 and 120578 at all nodesFigure 2(a) presents in the log-log scale the 119871

2norm of

the relative errors versus the number of nodes used indiscretizationWhen the spatial resolution goes down to 001that is 119873 = 100 the Euclidean norms of the relative errors

6 Journal of Computational Engineering

50 100 150 200N

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4L2

(rel

ativ

e err

or)

Conv rateu = 557

uxx = 610

(a)

50 100 150 200N

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

Conv rateu = 630

ux = 581

uxx = 484

120578 = 682

120578x = 537

dt = 10minus4

(b)

Figure 3 (a) Convergence rates for Compact Scheme III (b) Convergence rates for combined schemes in space with 119889119905 = 10minus4

drop to themachine zero in double precisionThe data pointsdemonstrate the sixth order convergence rate for both 120597119906120597119909and 120597120578119889119909

322 Compact Scheme II To see the performance of Com-pact Scheme II we use the known exact solution to computethe Euclidean norm of the relative errors for 12059721199061205971199092 at allnodes is In the log-log scale Figure 2(b) shows the 119871

2norm

of the relative error versus the number of nodes used in spaceBefore the spatial step reduces to 001 the 119871

2norm of the

relative errors of 12059721199061205971199092 drops to the machine zero Thisfigure clearly demonstrates the sixth order convergence rate

323 Compact Scheme III The performance of CompactScheme III is demonstrated below Taking advantage of theknown exact solution we compute the Euclidean normsof the relative errors for both 119906 and 120597

21199061205971199092 at all nodes

Figure 3(a) plots the 1198712norms of the relative errors versus

the number of nodes used in discretization The linearsystem generated from the Compact Scheme III is no longertridiagonal but it is still banded Because a fifth order schemeis used at the right end node 119894 = 119873 in order to adopt theNeumann boundary condition the convergence rates for 119906and 12059721199061205971199092 are slightly different but about sixth order

33 Convergence Rates of Combined Schemes in Space

331 Time Step 119889119905 = 10minus4 Nowwe examine the convergence

rates for all three compact schemes together over 1000 timesteps after the initial conditions are imposed During thisperiod of time Compact Schemes I II and III are used one

after another to solve these modified Boussinesq equationsFigure 3(b) presents the overall spatial convergence rates(indicated in the legend resp) for all variables after 1000 timesteps which was set to be 119889119905 = 10

minus4 The lowest convergencerate of 484 is observed for 12059721199061205971199092 because the variable 119906carries a second derivative and the boundary condition is notexplicitly given in terms of 12059721199061205971199092 The rest of the variableshave the sixth order convergence rate

332 Time Step 119889119905 = 10minus5 and 119889119905 = 10

minus6 Similar to Figures3(b) and 4(a) with 119889119905 = 10

minus5 and Figure 4(b) with 119889119905 = 10minus6

present the spatial convergence rates (indicated in the legend)involving Compact Schemes I II and III for all variables after1000 time steps The lowest convergence rate about 49 isexpected for 12059721199061205971199092 for the same reason as before For therest variables the sixth order convergence rates are clearlydemonstrated From Figures 3(b) 4(a) and 4(b) we observethat as the time step size reduces high order convergencerates are maintained for higher spatial resolution (larger119873 infigures)This is because a smaller time step size produces lesserror during time integration However as the spatial resolu-tion (119873) increases the rounding error increases accordinglyand eventually overtakes the truncation error Therefore wedo not see the errors keep decreasing as119873 increases

34 Convergence Rate in Time Integration After we exam-ined the spatial convergence rates now we demonstrate theconvergence rate in time integration To this end we fix thespatial step size to 119889119909 = 002 (119873 = 50) and vary the sizeof time step to plot the 119871

2norms of the relative errors in all

variables Figure 5(a) presents the rate of convergence versus

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Computational Engineering 5

x

Valu

e

1 125 15 175 2

4

8

12

16

20

24

28

ApproxuApproxuxApprox uxx

Exact uExact uxExact uxx

(a)

x

1 125 15 175 2

+ + + + ++

++

++

++

++

++

++

++

+

Valu

e

3

4

5

6

7

+Approx120578Exact 120578

Approx120578xExact 120578x

(b)

Figure 1 Comparison of the approximate and exact solutions for 119906 (a) and 120578 (b)

50 100 150 200

Conv rate

10minus13

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

N

ux = 623

120578x

ux

120578x

= 612

(a)

10minus12

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

L2

(rel

ativ

e err

or)

50 100 150 200N

Conv rate uxx = 593

(b)

Figure 2 Convergence rates for Compact Scheme I (a) and Compact Scheme II (b) in space

32 Convergence Rates of Separate Compact Schemes in Space

321 Compact Scheme I The performance of CompactScheme I is examined here Since the exact solutions areavailable we compute the Euclidean (119871

2) norms of the

relative errors for the first derivative of 119906 and 120578 at all nodesFigure 2(a) presents in the log-log scale the 119871

2norm of

the relative errors versus the number of nodes used indiscretizationWhen the spatial resolution goes down to 001that is 119873 = 100 the Euclidean norms of the relative errors

6 Journal of Computational Engineering

50 100 150 200N

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4L2

(rel

ativ

e err

or)

Conv rateu = 557

uxx = 610

(a)

50 100 150 200N

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

Conv rateu = 630

ux = 581

uxx = 484

120578 = 682

120578x = 537

dt = 10minus4

(b)

Figure 3 (a) Convergence rates for Compact Scheme III (b) Convergence rates for combined schemes in space with 119889119905 = 10minus4

drop to themachine zero in double precisionThe data pointsdemonstrate the sixth order convergence rate for both 120597119906120597119909and 120597120578119889119909

322 Compact Scheme II To see the performance of Com-pact Scheme II we use the known exact solution to computethe Euclidean norm of the relative errors for 12059721199061205971199092 at allnodes is In the log-log scale Figure 2(b) shows the 119871

2norm

of the relative error versus the number of nodes used in spaceBefore the spatial step reduces to 001 the 119871

2norm of the

relative errors of 12059721199061205971199092 drops to the machine zero Thisfigure clearly demonstrates the sixth order convergence rate

323 Compact Scheme III The performance of CompactScheme III is demonstrated below Taking advantage of theknown exact solution we compute the Euclidean normsof the relative errors for both 119906 and 120597

21199061205971199092 at all nodes

Figure 3(a) plots the 1198712norms of the relative errors versus

the number of nodes used in discretization The linearsystem generated from the Compact Scheme III is no longertridiagonal but it is still banded Because a fifth order schemeis used at the right end node 119894 = 119873 in order to adopt theNeumann boundary condition the convergence rates for 119906and 12059721199061205971199092 are slightly different but about sixth order

33 Convergence Rates of Combined Schemes in Space

331 Time Step 119889119905 = 10minus4 Nowwe examine the convergence

rates for all three compact schemes together over 1000 timesteps after the initial conditions are imposed During thisperiod of time Compact Schemes I II and III are used one

after another to solve these modified Boussinesq equationsFigure 3(b) presents the overall spatial convergence rates(indicated in the legend resp) for all variables after 1000 timesteps which was set to be 119889119905 = 10

minus4 The lowest convergencerate of 484 is observed for 12059721199061205971199092 because the variable 119906carries a second derivative and the boundary condition is notexplicitly given in terms of 12059721199061205971199092 The rest of the variableshave the sixth order convergence rate

332 Time Step 119889119905 = 10minus5 and 119889119905 = 10

minus6 Similar to Figures3(b) and 4(a) with 119889119905 = 10

minus5 and Figure 4(b) with 119889119905 = 10minus6

present the spatial convergence rates (indicated in the legend)involving Compact Schemes I II and III for all variables after1000 time steps The lowest convergence rate about 49 isexpected for 12059721199061205971199092 for the same reason as before For therest variables the sixth order convergence rates are clearlydemonstrated From Figures 3(b) 4(a) and 4(b) we observethat as the time step size reduces high order convergencerates are maintained for higher spatial resolution (larger119873 infigures)This is because a smaller time step size produces lesserror during time integration However as the spatial resolu-tion (119873) increases the rounding error increases accordinglyand eventually overtakes the truncation error Therefore wedo not see the errors keep decreasing as119873 increases

34 Convergence Rate in Time Integration After we exam-ined the spatial convergence rates now we demonstrate theconvergence rate in time integration To this end we fix thespatial step size to 119889119909 = 002 (119873 = 50) and vary the sizeof time step to plot the 119871

2norms of the relative errors in all

variables Figure 5(a) presents the rate of convergence versus

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Computational Engineering

50 100 150 200N

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4L2

(rel

ativ

e err

or)

Conv rateu = 557

uxx = 610

(a)

50 100 150 200N

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

Conv rateu = 630

ux = 581

uxx = 484

120578 = 682

120578x = 537

dt = 10minus4

(b)

Figure 3 (a) Convergence rates for Compact Scheme III (b) Convergence rates for combined schemes in space with 119889119905 = 10minus4

drop to themachine zero in double precisionThe data pointsdemonstrate the sixth order convergence rate for both 120597119906120597119909and 120597120578119889119909

322 Compact Scheme II To see the performance of Com-pact Scheme II we use the known exact solution to computethe Euclidean norm of the relative errors for 12059721199061205971199092 at allnodes is In the log-log scale Figure 2(b) shows the 119871

2norm

of the relative error versus the number of nodes used in spaceBefore the spatial step reduces to 001 the 119871

2norm of the

relative errors of 12059721199061205971199092 drops to the machine zero Thisfigure clearly demonstrates the sixth order convergence rate

323 Compact Scheme III The performance of CompactScheme III is demonstrated below Taking advantage of theknown exact solution we compute the Euclidean normsof the relative errors for both 119906 and 120597

21199061205971199092 at all nodes

Figure 3(a) plots the 1198712norms of the relative errors versus

the number of nodes used in discretization The linearsystem generated from the Compact Scheme III is no longertridiagonal but it is still banded Because a fifth order schemeis used at the right end node 119894 = 119873 in order to adopt theNeumann boundary condition the convergence rates for 119906and 12059721199061205971199092 are slightly different but about sixth order

33 Convergence Rates of Combined Schemes in Space

331 Time Step 119889119905 = 10minus4 Nowwe examine the convergence

rates for all three compact schemes together over 1000 timesteps after the initial conditions are imposed During thisperiod of time Compact Schemes I II and III are used one

after another to solve these modified Boussinesq equationsFigure 3(b) presents the overall spatial convergence rates(indicated in the legend resp) for all variables after 1000 timesteps which was set to be 119889119905 = 10

minus4 The lowest convergencerate of 484 is observed for 12059721199061205971199092 because the variable 119906carries a second derivative and the boundary condition is notexplicitly given in terms of 12059721199061205971199092 The rest of the variableshave the sixth order convergence rate

332 Time Step 119889119905 = 10minus5 and 119889119905 = 10

minus6 Similar to Figures3(b) and 4(a) with 119889119905 = 10

minus5 and Figure 4(b) with 119889119905 = 10minus6

present the spatial convergence rates (indicated in the legend)involving Compact Schemes I II and III for all variables after1000 time steps The lowest convergence rate about 49 isexpected for 12059721199061205971199092 for the same reason as before For therest variables the sixth order convergence rates are clearlydemonstrated From Figures 3(b) 4(a) and 4(b) we observethat as the time step size reduces high order convergencerates are maintained for higher spatial resolution (larger119873 infigures)This is because a smaller time step size produces lesserror during time integration However as the spatial resolu-tion (119873) increases the rounding error increases accordinglyand eventually overtakes the truncation error Therefore wedo not see the errors keep decreasing as119873 increases

34 Convergence Rate in Time Integration After we exam-ined the spatial convergence rates now we demonstrate theconvergence rate in time integration To this end we fix thespatial step size to 119889119909 = 002 (119873 = 50) and vary the sizeof time step to plot the 119871

2norms of the relative errors in all

variables Figure 5(a) presents the rate of convergence versus

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Computational Engineering 7

50 100 150 200N

10minus10

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus5

Conv rateu = 652

ux = 599

uxx = 489

120578 = 608

120578x = 540

(a)

50 100 150 200N

10minus10

10minus11

10minus9

10minus8

10minus7

10minus6

10minus5

L2

(rel

ativ

e err

or)

dt = 10minus6

Conv rateu = 629

ux = 597

uxx = 488

120578 = 581

120578x = 482

(b)

Figure 4 Convergence rates for combined schemes in space with (a) 119889119905 = 10minus5 and (b) 119889119905 = 10

minus6

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dx

dt

= 002

u

uxuxx

120578

120578x

(a)

dx = 0125

10minus12

10minus4 10minus5 10minus6

10minus11

10minus10

10minus9

10minus8

10minus7

10minus6

L2

(rel

ativ

e err

or)

dt

u

uxuxx

120578

120578x

(b)

Figure 5 Convergence rates for combined schemes in time1198712normof the relative errors versus the size of time step at fixed spatial resolution

(a) 119889119909 = 002 and (b) 119889119909 = 00125

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Computational Engineering

Time0 05 1 15

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus5

u

120578

dx = 002 dt =

n steps = 168000

(a)

Time0 05 1 15

10minus12

10minus11

10minus10

10minus9

10minus8

L2

(rel

ativ

e err

or)

u

uxx

10minus5dx = 002 dt =

n steps = 168000

120578x

(b)

Figure 6 Long time integration over 168000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus5

the size of time step for all variables A second order timeconvergence is observed when 119889119905 is less than 10

minus5 Sincethe spatial step size is still large at 119889119905 = 10

minus5 the timeconvergence rate is impaired especially for 119906

119909119909 After we

reduce 119889119909 to 00125 the second order time convergence rateis restored upto 119889119905 = 10

minus5 for all variablesWhen 119889119905 is furtherdiminished to 10minus6 the time convergence rate becomes lesssensitive to 119889119905 again but the 119871

2norm of the relative error

of 119906119909119909

is still better than the case with a larger spatial step119889119909 = 002

35 The Effect of Long Time Integration To examine theeffect of long time integration we fix the spatial resolutionto 119873 = 50 that is 119889119909 = 002 and use 119889119905 = 10

minus5 to plotthe overall convergence rates for all variables with CompactSchemes I II III implemented one after another duringcomputations Since all three schemes are of sixth orderaccuracy the magnitude of discretization error is (002)6 =64 times 10

minus11 the time step size has to be small enough so thatthe time integration errorwill be of the same order as in spaceFigure 6(a) shows the evolution of convergence rates for 119906and 120578 over 168000 steps in time integration with 119889119905 = 10

minus5It is observed that during this long time integration the 119871

2

norms of the relative errors in 119906 and 120578 are bounded below10minus9 although slightly growing over time Figure 6(b) shows

the evolution of convergence rates for the derivative of 119906 and120578 in the same run The 119871

2norms of the relative errors in

119906119909 119906119909119909 and 120578

119909are below 10

minus8 The 1198712norm of the relative

error in 119906119909119909

is almost one order higher than those in 119906119909and

120578119909 The reason is that although all derivatives of 119906 and 120578

are treated as the state variables just like 119906 and 120578 themselvesand solved in Compact Schemes I II III simultaneously no

Dirichlet boundary condition for 119906119909119909

is specified and 119906119909119909

issolved after the time integration therefore the relative errorin 119906119909119909

is superposed by the time integration errorNext we reduce the time step size to119889119905 = 10

minus6 Figure 7(a)shows the evolution of convergence rates for 119906 and 120578 aftera time integration period over 220000 steps It is observedthat during this time period the 119871

2norms of the relative

errors in 119906 and 120578 are well bounded below 10minus10 one order

lower than in Figure 6(a) Figure 7(b) presents the evolutionof convergence rates for the derivatives of 119906 and 120578 in the samerunThe119871

2norms of the relative errors in 119906

119909and 120578119909are below

10minus10 while the relative error in 119906

119909119909is stable around 10

minus9There is no indication of error growth since the time step sizeis small enough to stabilize the growth of the time integrationerror

4 Conclusion

In this paper coupled compact methods are used to solvenonlinear differential equations with third order mixedderivatives in space and time We have developed sixthorder compact schemes for numerical solutions to modifiedBoussinesq-type of equations a coupled system of nonlineardifferential equations with strong convective and dispersiveterms The novelty of this work is summarized as belowFirst the state variables include the original unknowns 119906120578 and their first and second derivatives However all statevariables are not solved at once because solving 119906 119906

119909 119906119909119909 120578

and 120578119909simultaneously in one linear system requires to triple

the size of the linear system for solving 119906 and 120578 togetherTo reduce memory usage for potentially very large systemswe presolve the first and the second derivatives of 119906 in

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Computational Engineering 9

10minus12

10minus11

10minus10

10minus9L2

(rel

ativ

e err

or)

u

120578

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

(a)

10minus12

10minus11

10minus10

10minus9

L2

(rel

ativ

e err

or)

10minus6dx = 002 dt =

n steps = 220000

Time

0 005 01 015 02

u

uxx120578x

(b)

Figure 7 Long time integration over 220000 time steps for 119906 and 120578 (a) and for the derivatives of 119906 and 120578 (b) the time step size being 10minus6

tridiagonal systems After the time integration we solve for119906 and 119906

119909119909 In other words 119906 120578 and their derivatives are

solved in two consecutive time levels Second the coupledpartial differential equations are solved in space and timealternately This is especially effective for strong nonlinearterms No explicit linearization is needed Third for thenonlinear partial differential equations with mixed thirdorder derivative the implicit differencing is applied in spaceand explicit differencing is used in time integration Thisunique treatment avoids the linearization of any nonlinearterms This could be very useful for strongly nonlinear andtime-dependent partial differential equations

With the aid of convergence analyses in space andtime by fixing either the time step or the spatial step theexpected accuracy of these schemes is demonstrated Longtime integrations were performed to show the stability ofthese schemes

In the subsequent work we investigate the effectivenessof the proposed schemes for solutions with oscillatory signssuch as the velocity of water waves Upwind treatment will beadopted for strong hyperbolic systems

Acknowledgments

This work was supported by National Science Foundationunder grants DMS-1115546 and DMS-1115527 and the stateof Louisiana Board of Regents under grant LEQSF (2007-10)-RD-A-22 The authors also appreciate their colleague DrBernd Schroeder for providing constructive suggestions

References

[1] H O Kreiss ldquoMethods for the approximate solution of timedependent problemsrdquo GARP Report No 13 1975

[2] R S Hirsh ldquoHigher order accurate difference solutions offluidmechanics problems by a compact differencing techniquerdquoJournal of Computational Physics vol 19 no 1 pp 90ndash109 1975

[3] Y Adam ldquoHighly accurate compact implicit methods andboundary conditionsrdquo Journal of Computational Physics vol 24no 1 pp 10ndash22 1977

[4] J D Hoffman ldquoRelationship between the truncation errorsof centered finite-difference approximations on uniform andnonuniformmeshesrdquo Journal of Computational Physics vol 46no 3 pp 469ndash474 1982

[5] S C R Dennis and J D Hudson ldquoCompact h4 finite-differenceapproximations to operators of Navier-Stokes typerdquo Journal ofComputational Physics vol 85 no 2 pp 390ndash416 1989

[6] M M Rai and P Moin ldquoDirect simulations of turbulentflow using finite-difference schemesrdquo Journal of ComputationalPhysics vol 96 no 1 pp 15ndash53 1991

[7] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992

[8] S-T Yu L S Hultgren and N-S Liu ldquoDirect calculationsof waves in fluid flows using high-order compact differenceschemerdquo AIAA Journal vol 32 no 9 pp 1766ndash1773 1994

[9] K Mahesh ldquoA family of high order finite difference schemeswith good spectral resolutionrdquo Journal of ComputationalPhysics vol 145 no 1 pp 332ndash358 1998

[10] W Dai and R Nassar ldquoA compact finite difference scheme forsolving a three-dimensional heat transport equation in a thinfilmrdquo Numerical Methods for Partial Differential Equations vol16 no 5 pp 441ndash458 2000

[11] W Dai and R Nassar ldquoA compact finite-difference schemefor solving a one-dimensional heat transport equation at themicroscalerdquo Journal of Computational and Applied Mathemat-ics vol 132 no 2 pp 431ndash441 2001

[12] H Sun and J Zhang ldquoA high-order finite difference discretiza-tion strategy based on extrapolation for convection diffusion

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Computational Engineering

equationsrdquo Numerical Methods for Partial Differential Equa-tions vol 20 no 1 pp 18ndash32 2004

[13] J A Ekaterinaris ldquoImplicit high-resolution compact schemesfor gas dynamics and aeroacousticsrdquo Journal of ComputationalPhysics vol 156 no 2 pp 272ndash299 1999

[14] M Li T Tang and B Fornberg ldquoA compact fourth-orderfinite difference scheme for the steady incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 10 pp 1137ndash1151 1995

[15] J S Shang ldquoHigh-order compact-difference schemes fortime-dependent Maxwell equationsrdquo Journal of ComputationalPhysics vol 153 no 2 pp 312ndash333 1999

[16] W F Spotz and G F Carey ldquoExtension of high-order compactschemes to time-dependent problemsrdquo Numerical Methods forPartial Differential Equations vol 17 no 6 pp 657ndash672 2001

[17] H El-Zoheiry ldquoNumerical study of the improved Boussinesqequationrdquo Chaos Solitons and Fractals vol 14 no 3 pp 377ndash384 2002

[18] J Zhang H Sun and J J Zhao ldquoHigh order compact schemewith multigrid local mesh refinement procedure for convectiondiffusion problemsrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 191 no 41-42 pp 4661ndash46742002

[19] T K Sengupta G Ganeriwal and S De ldquoAnalysis of central andupwind compact schemesrdquo Journal of Computational Physicsvol 192 no 2 pp 677ndash694 2003

[20] M Y Shen Z B Zhang and X L Niu ldquoA new way for con-structing high accuracy shock-capturing generalized compactdifference schemesrdquo Journal of Computer Methods in AppliedMechanics and Engineering vol 192 no 25 pp 2703ndash27252003

[21] A Jocksch N A Adams and L Kleiser ldquoAn asymptoticallystable compact upwind-biased finite-difference scheme forhyperbolic systemsrdquo Journal of Computational Physics vol 208no 2 pp 435ndash454 2005

[22] M A H Mohamad S Basri I Rahim and M Mohamad ldquoAnAUSM-based third-order compact scheme for solving Eulerequationsrdquo International Journal of Engineering amp Technologyvol 3 no 1 pp 1ndash12 2006

[23] S L Anthonio and K R Hall ldquoHigh-order compact numericalschemes for non-hydrostatic free surface flowsrdquo InternationalJournal for NumericalMethods in Fluids vol 52 no 12 pp 1315ndash1337 2006

[24] E C Cruz and Q Chen ldquoNumerical modeling of nonlinearwater waves over heterogeneous porous bedsrdquo Ocean Engineer-ing vol 34 no 8-9 pp 1303ndash1321 2007

[25] P H Chiu and T W H Sheu ldquoOn the development of adispersion-relation-preserving dual-compact upwind schemefor convection-diffusion equationrdquo Journal of ComputationalPhysics vol 228 no 10 pp 3640ndash3655 2009

[26] Q Zhou Z Yao F He and M Y Shen ldquoA new family of high-order compact upwind difference schemes with good spectralresolutionrdquo Journal of Computational Physics vol 227 no 2 pp1306ndash1339 2007

[27] Y V S S Sanyasiraju and N Mishra ldquoSpectral resolutionedexponential compact higher order scheme (SRECHOS) forconvection-diffusion equationsrdquo Journal of Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4737ndash4744 2008

[28] M Dehghan and A Mohebbi ldquoHigh-order compact boundaryvalue method for the solution of unsteady convection-diffusion

problemsrdquo Mathematics and Computers in Simulation vol 79no 3 pp 683ndash699 2008

[29] A Shah H Guo and L Yuan ldquoA third-order upwind compactscheme on curvilinear meshes for the incompressible Navier-Stokes equationsrdquo Communications in Computational Physicsvol 5 no 2-4 pp 712ndash729 2009

[30] S-S Xie G-X Li and S Yi ldquoCompact finite difference schemeswith high accuracy for one-dimensional nonlinear Schrodingerequationrdquo Journal of Computer Methods in Applied Mechanicsand Engineering vol 198 no 9-12 pp 1052ndash1060 2009

[31] MTonelli andMPetti ldquoHybrid finite volumemdashfinite differencescheme for 2DH improved Boussinesq equationsrdquo CoastalEngineering vol 56 no 5-6 pp 609ndash620 2009

[32] R L Burden and J D Faires Numerical Analysis ThomsonBrooksCole Belmont Calif USA 8th edition 2005

[33] D Liu W Kuang and A Tangborn ldquoHigh-order compactimplicit difference methods for parabolic equations in geody-namo simulationrdquo Advances in Mathematical Physics vol 2009Article ID 568296 23 pages 2009

[34] L Gamet F Ducros F Nicoud and T Poinsot ldquoCompactfinite difference schemes on nonuniform meshes applicationto direct numerical simulations of compressible flowsrdquo Inter-national Journal for Numerical Methods in Fluids vol 29 pp159ndash191 1999

[35] O V Vasilyev ldquoHigh order finite difference schemes on non-uniform meshes with good conservation propertiesrdquo Journal ofComputational Physics vol 157 pp 746ndash761 2000

[36] M Hermanns and J A Hernandez ldquoStable high-order finite-difference methods based on non-uniform grid point distribu-tionsrdquo International Journal for Numerical Methods in Fluidsvol 56 no 3 pp 233ndash255 2008

[37] M W Dingemans Water Wave Propagation over UnevenBottoms World Scientific 1997

[38] B Gustafsson H Kreiss and J Oliger Time Dependent Prob-lems and Difference Methods John Wiley amp Sons 1995

[39] D Gottlieb and S A Orszag Numerical Analysis of SpectralMethods Theory and Applications Society for Industrial andApplied Mathematics 1977

[40] C Pozrikidis Introduction to Finite and Spectal ElementMethodUsing MATLAB Chapman amp HallCRC 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of