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ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2007, Vol. 47, No. 12, pp. 19701989. Pleiades Publishing, Ltd., 2007.Original Russian Text M.V. Popov, S.D. Ustyugov, 2007, published in Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 12, pp. 20552075.

Piecewise Parabolic Method on Local Stencilfor Gasdynamic Simulations

M. V. Popov and S. D. Ustyugov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences,Miusskaya pl. 4, Moscow, 125047 Russia

e-mail: [email protected], [email protected]

Received May 30, 2007; in final form, June 5, 2007

Abstract

A numerical method based on piecewise parabolic difference approximations is proposedfor solving hyperbolic systems of equations. The design of its numerical scheme is based on the conser-vation of Riemann invariants along the characteristic curves of a system of equations, which makes itpossible to discard the four-point interpolation procedure used in the standard piecewise parabolicmethod (PPM) and to use the data from the previous time level in the reconstruction of the solutioninside difference cells. As a result, discontinuous solutions can be accurately represented without addingexcessive dissipation. A local stencil is also convenient for computations on adaptive meshes. The new

method is compared with PPM by solving test problems for the linear advection equation and the invis-cid Burgers equation. The efficiency of the methods is compared in terms of errors in various norms. Atechnique for solving the gas dynamics equations is described and tested for several one- and two-dimensional problems.

DOI: 10.1134/S0965542507120081

Keywords:

numerical methods in gas dynamics, local stencil, Riemann invariants, numerical methods,hyperbolic systems of equations, PPM, PPML

1. INTRODUCTION

The piecewise parabolic method (PPM) for the numerical solution to hyperbolic systems of equationswas first proposed in [1] and was found to be efficient in numerical practice. The method is third-order accu-rate in space and second-order accurate in time. To determine boundary points in the construction of a parab-ola in each difference cell, the PPM method reconstructs the variables on an extended four-point stencil.This leads to excessive dissipation in the scheme, for example, on contact discontinuities. As a result, a spe-cial adjustable technique is required for steepening the discontinuity fronts. Additionally, computational dif-ficulties arise in the construction and use of well-defined boundary conditions in the computational domain.The application of high-order interpolation on an extended stencil ensures high-quality results for smoothsolutions but leads to noticeable oscillations at discontinuities. The situation can be improved by applyinglimiters. In that case, however, difficulties are encountered in problems with oscillatory solutions, since theoscillation amplitude is damped due to the strong diffusion caused by limiters in the neighborhood of steepfronts. Another shortcoming of limiters as applied to the preservation of discontinuity profiles is that theystrengthen short oscillation modes, which increase instability when the original equations involve disper-sion terms.

We proposed a piecewise parabolic method on a local stencil in which the boundary points of each parab-ola are determined using data from the previous time level according to the method of characteristics. A sim-

ilar idea was first used in [2] to construct a first-order-accurate difference scheme inside difference cells andwas discussed in [3]. In this approach, discontinuous solutions can be accurately represented without addingexcessive dissipation inherent in schemes on extended stencils. The piecewise parabolic method on a localstencil will be referred to as PPML.

A widely used modern practice is that difference schemes are tested in numerical experiments and theirerrors are compared in different norms (see [47]). We compare the PPM and PPML methods by numeri-cally solving the Cauchy problem for the linear advection equation and the nonlinear inviscid Burgers equa-tion. The analysis is based on the technique used in [57] in a similar study of various difference schemes.The PPM method was tested and compared with other difference schemes in [7], where it was shown thatPPM is one of the best modern techniques for computing the gas dynamics equations.


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No. 12

2007

PIECEWISE PARABOLIC METHOD 1971

In this paper, we describe a numerical scheme for gasdynamic simulation based on the new method. Thegas dynamics equations have three characteristics along which Riemann invariants interpreted as waveamplitudes are conserved. The equations are nonlinear, and a local basis of eigenvectors has to be deter-mined in each cell in order to apply characteristic analysis. The nonlinear problem of determining fluxes atthe cell boundaries can then be linearized in the neighborhood of each cell boundary, which makes it possi-ble to use a piecewise parabolic distribution of physical variables.

2. PIECEWISE PARABOLIC METHOD (PPM)Consider a homogeneous one-dimensional mesh of size h

with the desired function q

(

x

) defined on it.Let q

i

and q

i

+ 1/2

denote the values at the cell centers and on the boundary, respectively. According to thePPM method, q

(

x

) inside each cell can be approximated by a parabola (see Fig. 2):

(1)

where

Formula (1) satisfies the relation

In domains where q

(

x

) is smooth and has no extrema, its boundary value belongs to the interval

(2)

In this case, we have = = q

i

+ 1/2

and = = q

i

1/2

. The values q

i

+ 1/2

are computed using the

fourth-order interpolation procedure

(3)

where

For the solution to be monotone and condition (2) to be satisfied, the values

q

i

in (3) have to be replacedwith

The values and have to be updated in domains where the solution q

(

x

) is not monotone. If q

i

is a

local maximum or minimum, then interpolation function (1) has to be a constant; i.e., = = q

i

. If q

i

is

too close to or , then parabola (1) can have an extremum inside a cell (here, |

q

i

|

< ). In this case,

q x( ) qiL qi qi

6( )1 ( )+( ),+=

x xi 1/2( )h1, qi qi

Rqi

L,= =

qi6( )

6 qi 1/2( ) qiL

qiR

+( )( ).=

qi h1

q x( ) x.dxi 1/2

xi 1/2+

=

qi 1/2+ qiqi 1+[ ].

qiR

qi 1+L

qiL

qi 1R

qi 1/2+ 1/2( ) qi qi 1++( ) 1/6( ) qi 1+ qi( ),=

qi 1/2( ) qi 1+ qi 1+( ).=

mqimin qi 2 qi 1+ qi 2 qi qi 1, ,( ) qi( ), qi 1+ qi( ) qi qi 1( )sgn 0,>

0, qi 1+ qi( ) qi qi 1( ) 0.

=

qiL

qiR

qiL

qiR

qi

Lq

i

Rq

i

6( )

qLi

qRi

qi

xi 1/2 xi + 1/2xi

Fig. 1.


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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 47 No. 12 2007

POPOV, USTYUGOV

and are chosen so that the extremum is shifted toward the cell boundary. These conditions can be

written as

(4)

and

(5)

After q(x) has been defined, we can calculate its mean over the interval [xi+ 1/2yxi+ 1/2] (fory> 0):

(6)

Consider the linear advection equation

(7)

When a discontinuity decays on the boundary of two adjacent cells at the point xi+ 1/2, there arises a mean

state q*(xi+ 1/2, t). The advection equation has a single characteristic determined by the condition dx/dt= a.Therefore, for a>0, the solution at the time t= t0+is defined as an average over the spatial interval [xi+ 1/2

axi+ 1/2]; i.e., q*(xi+ 1/2, t0+ ) = = (a). For a< 0, the determining interval (region of influ-

ence) is [xi+ 1/2xi+ 1/2+ a]. In this case, we have q*(xi+ 1/2, t0+ ) = = (a), where

(8)

Here,y> 0. The flux on the boundary of adjacent cells in the Riemann problem is given by the formula

For convenience, we can introduce the functions a+= max(a, 0) = (a+ |a |)/2 and a= min(a, 0) = (a |a |)/2.Then

(9)

The values for a< 0 and for a> 0 can be arbitrary.

3. PIECEWISE PARABOLIC METHOD ON A LOCAL STENCIL (PPML)

A shortcoming of the PPM method is that it uses four-point interpolation procedure (3), which smoothsthe discontinuous solutions q(x), for example, at shock fronts or contact discontinuities. Instead of the inter-

polation procedure, we suggest that qi+ 1/2, for example, on the right cell boundary be determined by trans-ferring the parabola value from the preceding time step along the characteristic dx/dt= a. In other words,qi+ 1/2at the time t= t0+ is calculated by moving along the characteristic from the pointxi+ 1/2on the rightcell boundary with the value qi+ 1/2up to the time t= t0(see Fig. 2).

Therefore, for a> 0, we have

(10)

where = (x xi 1/2)h1= (h a)h1= 1 ah1. All the values on the right-hand side of (10) are takenfrom the previous time step t= t0 . For a< 0, the value qi+ 1/2is determined from the parabola in the cell

qiL

qiR

qiL

qi, qiR

qi, if qiL

qi( ) qi qiR

( ) 0,= =

qiL

3qi 2qiR

, if qi qi6( ) qi( )

2,>=

qiR

3qi 2qiL

, if qi qi6( ) qi( )

2.


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indexed by i + 1:

where = ah1.Thus, an approximating parabola is constructed in each cell. After verifying conditions (4) and (5), we

use formulas (6) and (8) to calculate (a) for a> 0 or (a) for a < 0 and then determine fluxes(9).

In this modification, the computational algorithm is implemented on a local stencil, since the boundarypoints of the piecewise parabola at the subsequent time step are determined without using the solution inneighboring cells.

4. TESTING THE METHODS BY SOLVING THE ADVECTION EQUATION

The methods were tested according to the technique described in [6]. We considered the Cauchy problemfor linear advection equation (7) with the initial conditions q(x, 0) = 0 forx (, l1) (l2, +)and q(x,0) = q0(x) forx [l1, l2]. Various profiles were used as q0(x):

(left triangle),

rectangle),

(cosine),

qi 1/2+ t0 +( ) qi 1+L

t0 +( ) qi 1+L qi 1+ qi 1+

6( )1 ( )+( ),+= =

qi 1/2+L

qi 1/2+R

q0 x( )1

l2 l1------------- x l1( )=

q0 x( ) 1=

q0 x( )1

2---

1

2---

2l2 l1------------- x l1( )

cos=

q0 x( )

2

3 l11 l1( )----------------------- x l1( ) 1, x l1 l11 ),,[+

1

3---, x l11 l22,[ ],2

3 l2 l22( )----------------------- x l2( ) 1, x l22 l2 ], (tooth),,(+

=

q0 x( )

2

3 l12 l1( )----------------------- x l1( ) 1, x l1 l12 ),,[+

2

3 l2 l12( )----------------------- x l2( ) 1, x l12 l2,[ ], (M),+

=

t0+

t0

xi 1/2 x xi + 1/2

qi + 1/2

qi

a x

Fig. 2.

t

.

.

.

.

.

.

.


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POPOV, USTYUGOV

(right triangle).

The numerical solution to Eq. (7) was compared with the exact one given by the formula

For this purpose, we calculated the norms in C,L1,L2 , and on = (, +) [0, T] (integrated overtime):

The following parameters were used in the computations: l= 520, l1= 10, l2= 30, l11= 16 , l22= 23 ,

l12= 20, T= 400, h= 1, anda= 1. For these values of h and a, the Courant number coincides with thetime step . Over the time T = 400, the profile travels over 20 of its own lengths, which are sufficient to exam-ine the properties of the numerical scheme.

Finite-difference schemes were analyzed in [6]. Values between grid nodes are not defined in suchschemes, and norms have to be calculated using their finite-difference analogues, in which the integrals arereplaced by sums over grid nodes and time steps. In the PPM method, the solution inside difference cells isapproximated by a parabola; therefore, the spatial integrals in the formulas for theL1andL2norms are pre-cisely calculated.

To calculate the norm in C, each cell indexed by i was partitioned byM= 200 nodes (indexed byj) at

which the values at the time kwere determined:

Here,Nis the number of cells and Kis the number of time steps (k = Kcorresponds to t = T).

While calculating the norm in , it should be borne in mind that all the initial profiles q0(x) (except for

the cosine one) involve discontinuity points at which = . Therefore, the evaluation of exact integrals

makes no sense. Instead, we use a finite-difference analogue of with the norm defined as in [6]:

Obviously, the values of the norm in then depend on the degree of detail of the grid and tend to ash 0 for profiles involving discontinuity points.

Figure 3 shows the numerical solutions to Eq. (7) at times tfrom 0 to Tfor three of the six initial profilesq0(x) at the Courant number = 0.8. Using these results, we can compare the accuracy of the profiles trans-ferred in the PPM and PPML methods. Tables 1 and 2 present the time-integral errors that are the norms ofthe difference between the exact and numerical solutions for PPM and PPML, respectively. It can be seenthat PPML produces better solutions for all the profiles due to lower dissipation.

q0 x( )1

l2 l1------------- l2 x( )=

q0 x( )

0, x at l1,

=

W21

f C f , f L1max f x t, f L2dd

+

0

T

f2

x tdd

+

0

T

1/2

,= = =

fW2

1 fx'( )2

x tdd

+

0

T

1/2

.=

2

3---

1

3---

qi j,k

q C qi j,k

.0 i N 0 j M 0 k K , ,

max=

W21

qx'

W21

qW

2

1 h1

qi 1k

qik

( )2

k 0=

K

i 0=

N

1/2

.=

W21


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5. TESTING THE METHODS BY SOLVING THE BURGERS EQUATION

The PPM and PPML methods were also used to solve the Cauchy problem for the inviscid Burgers equa-tion, which describes the origin and propagation of shock waves. The arising shock waves are similar tothose in gas dynamics problems. The Burgers equation has the form

or

(11)

Obviously, the characteristic curve of Eq. (11) is determined by the condition dx/dt= q. Since the charac-teristic velocity qis not a constant, Eq. (11) is nonlinear. The velocity of propagation at each point describedby the wave equation can be different, which leads to the formation of discontinuous solutions and shockwaves. In certain cases, the analytical solution to Eq. (11) can be written as

(12)

where q(x, 0) = q0(x) is the initial profile. Expression (12) holds true in the case of a smooth initial profileuntil shock waves appear, after which the solution is represented as a piecewise linear function.

qt------

x------

q2

2-----

+ 0,=

qt qqx+ 0.=

q x t,( ) q0 x qt( ),=

Fig. 3.

PPM PPML

Table 1

PPM Left triangle Rectangle Cosine Tooth M Right triangle

C 0.66672 0.64466 0.065666 0.68878 0.67407 0.66621

L1 448.1290 714.6680 101.9580 1001.1000 998.2290 473.5950

L2 10.8551 14.7612 1.7026 16.5501 15.8830 10.9834

15.5816 22.0077 1.5848 22.8209 22.4006 15.6465W21

Table 2

PPML Left triangle Rectangle Cosine Tooth M Right triangle

C 0.61997 0.61360 0.040749 0.62633 0.62209 0.63704

L1 363.3940 625.4640 39.4735 783.3780 790.9780 365.4070

L2 9.9448 13.7838 0.79444 14.7433 14.4267 10.0368

14.9228 21.0576 0.81280 21.6119 21.3418 14.9330W21


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POPOV, USTYUGOV

Since the flux in the Burgers equation is F= q2/2, relation (9) is replaced by

where (y) and (y) are given by formulas (6) and (8), respectively, and ai+ 1/2is the velocity on

the boundary of adjacent cells, which is calculated by averaging the right and left boundary states:

We employ one of the test problems from [7] that demonstrates the evolution of a discontinuous profileinvolving the collision of shock waves and the expansion of rarefaction regions.

The computational domain was divided intoN= 50 cells, and the computations were performed for =0.4. The time step was determined by the Courant condition

The initial profile of the solution was specified as

Figure 4 shows the analytical solution to Eq. (11) for this initial profile at t= 0, 0.6, and 2.0. Initially,there are two shock waves at x= 2 and x= 3 that propagate toward each other and there is a rightward-expanding discontinuity atx= 0.2 and a leftward-expanding discontinuity atx = 4.8. It can be seen that, overtime, two expanding fans appear atx = 0.2 and 4.8. At t= 1, the shock waves collide and merge into a singleone moving to the left.

Figure 5 displays the numerical solutions (dots) produced by PPM at t = 0.6 and 2.0. The exact solutionis depicted by the solid line. The numerical results obtained with PPML at the same times are shown inFig. 6. The arrows indicate the areas where the methods exhibit visible differences. Specifically, in these

Fi 1/2+

1

2--- qi 1/2+

Lai 1/2+ ( )( )

2, ai 1/2+ 0,

1

2--- qi 1/2+

Rai 1/2+ ( )( )

2, ai 1/2+ 0,


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areas, the PPML solution (Fig. 6) exhibits smaller deviations from the exact one than the PPM solution(Fig. 5). The expansion regions observed in this test are similar to rarefaction waves occurring in gas dynam-ics problems. This means that the solutions produced by PPML are more accurate in areas where rarefactionwaves develop. Table 3 lists the errors inL1for variousNat t= 0.6 and 2.0. It can be seen that PPML yieldsconsiderably better results.

6. GAS DYNAMICS EQUATIONS

Consider the conservative form of the gas dynamics equations in two dimensions:

(13)

where

tU xF yG+ + 0,=

U

uvE

, F

u

u2 p+uv

E p+( )u

, G

vuv

v2 p+E p+( )v

.= = =

1.0

0.5

0

0.5

1.0

qt= 0.6

543210

t= 2.0

543210

x

Fig. 5.

1.0

0.5

0

0.5

1.0

q

543210

t = 0.6

543210

t = 2.0

xFig. 6.


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POPOV, USTYUGOV

System (13) is supplemented by the total energy equation

(14)

and the ideal-gas equation of state

(15)

where is the specific internal energy.The gas dynamics equations are solved using the conservative difference scheme

(16)

The half-integer spatial indices at the fluxes show the cell boundaries to which they correspond. The half-integer time step n+ 1/2 means that we use the fluxes averaged over the time step , which gives second-order accuracy in time.

The solution inside each cell is approximated by a parabola along each coordinate axis. The boundarypoints of each piecewise parabola are determined by the property that the Riemann invariants are conservedalong the characteristics of the original linearized system. To compute the fluxes at the cell boundaries, thegeneralized Riemann problem with an arbitrary distribution of the desired function inside a cell is trans-formed into an equivalent Riemann problem with this distribution specified by a piecewise constant func-

tion.In PPM, a piecewise parabola in each cell is constructed using the physical variables V= (, u, v,p).

For this reason, to determine the fluxes, the gas dynamics equations are additionally used in the nonconser-vative form

(17)

The matrices Aand Bcan be calculated via the Jacobians of the conservative system and the transitionmatrix

(18)

For example, forA, we have

The matricesAandBhave a complete set of right and left eigenvectors corresponding to real eigenvalues(the gas dynamics equations are hyperbolic). Therefore,AandBcan be decomposed in terms of their eigen-vectors. For example, forA(which corresponds to thexdirection), we have

(19)

Here, the columns ofRxare the right eigenvectors (p= 1, , 4);Lxis the inverse ofRx with its rows being

E u2

v

2+( )2

--------------------------+=

p 1( ),=

Ui j,n 1+

Ui j,n

x------ Fi 1/2+ j,

n 1/2+Fi 1/2 j,

n 1/2+( )

y------ Gi j, 1/2+

n 1/2+Gi j, 1/2

n 1/2+( ).=

tV AxV ByV+ + 0.=

MUV-------.=

A M1 FU-------M.=

A RxxLx.=

rxp

Table 3

Nt= 0.6 t= 2.0

PPM PPML PPM PPML

64 8.60 102 4.27 102 1.40 101 7.41 102

128 7.49 102 5.86 102 9.00 102 4.95 102

256 4.75 102 2.33 102 5.84 102 2.02 102

512 3.18 102 1.39 102 3.37 102 8.99 103

1024 1.55 102 5.81 103 1.89 102 7.12 103


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the left eigenvectors :

and xis a diagonal eigenvalue matrix: (x)ij= 0 for ij and ((x)ij= p for i=j=p, where pis the solutionto the characteristic equation

(20)

Equation (20) is solved by the eigenvalues 1= u+ c, 2= 3= u, and 4= u c, where cis the speed ofsound: c2= (p)s= (p)+ (p)p/2.

To construct piecewise parabolas at the next time step, we have to determine the values (states) at the cellboundaries and the values at the cell centers.

6.1. Computation of the Boundary Values of Piecewise Parabolas

For simplicity, Eqs. (17) are replaced by the one-dimensional system

(21)

(the two-dimensional case will be considered later). Plugging (19) into (21) and multiplyingLon the left,we obtain

(22)

The vector V(x, t) is decomposed in the local basis of right eigenvectors rpfixed for a given cell:

(23)

Substituting (23) into (22) yields

(24)

Equations (24) mean that the coefficients p(x, t) in (23) (which correspond to wave amplitudes) are con-served along each characteristicxp(t) defined by the condition

i.e., they are Riemann invariants. The value of a Riemann invariant on the cell boundary atx=xi+ 1/2at thetime t+ can be calculated in terms of its value at t:

(25)

As an example, Fig. 7 shows two adjacent cells indexed by iand i + 1. The set of characteristics is

denoted byp1in the cell iand byp2in the cell i + 1. One of the characteristics (t) with the eigenvalue

> 0 is shown in the cell i. The characteristic (t) corresponding to < 0 is shown in the cell i+ 1.

According to (25), the amplitude at point 3for the wave propagating in the cell ialong the characteristic

(t) corresponding to the eigenvalue is determined by the wave amplitude at point 1. Similarly, the

amplitude at point 3for the wave propagating in the cell i+ 1 along the characteristic (t) with the eigen-

value is determined by the wave amplitude at point 2.

The state at point 3calculated via formula (23) by summing up values over the eigenvectors fixed in the

cell iand associated with positive eigenvalues (i.e., over the indexp1for all > 0) is the left state with

respect to the boundary and is denoted by VL. Analogously, the state at point 3calculated by summing up

lxp

Rx

1 0 1 1

c/ 0 0 c/0 1/ 0 0

c2

0 0 c( )2

, Lx

0 /(2c) 0 1/(2c2)0 0 0

1 0 0 1/ c2

0 /(2c) 0 1/(2c2)

;= =

det E A 0.=

tV x t,( ) AxV x t,( )+ 0=

LtV LxV+ 0.=

V x t,( ) p x t,( )rp.p

=

tp px

p+ 0, p 1 4., ,= =

dxp/dt p,=

p xi 1/2+ t +,( ) p

xi 1/2+ p t,( ).=

xp1

p1

xp2

p2

xp1

p1

xp2

p2

p1


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POPOV, USTYUGOV

values over the eigenvectors in the cell i + 1 (i.e., over the indexp2for all < 0) is the right state with

respect to the boundary and is denoted by VR.

Figure 8 displays a piecewise parabolic approximation to one of the components V(x) of the vector func-tion V(x, t) at some time inside cells. The arrows indicate the values to the left and right of the boundary;here, VLVR. The dotted lines show the mean values for each cell. For example, the mean for the cell i is

The states VLand VRare required for constructing piecewise parabolas at the next time step. In the stan-dard PPM method, these states are matched so that

(26)

In PPM, condition (26) holds by construction, since Vi+ 1/2is calculated using the fourth-order interpolationprocedure on a four-point stencil. In the case under study, the parabolas can be matched by solving the Rie-mann problem, for example, by Roes method (see [8]). To do this, we proceed from the simple to conser-vative variables:

(27)

where = are the right eigenvectors for conservative system (13) (Mis the transition matrix given

by (18)). The values ULand URare calculated according to the eigenvector basis chosen in the cellsiandi+ 1. Numerical computations have shown that the results are hardly affected by the choice of a basis. The

values of are calculated from the state V*determined by the formulas

(28)

(see [8]). Here, uis a velocity component and h= (E+p)/ is the specific enthalpy.

The eigenvalues *phave the form *1= u* + c*, *2= *3= u*, and *4= u* c*, where c* =(p* =p(*, *)). The amplitudes *pare calculated from the state V*according to (23):

When the PPML method is applied to two-dimensional problems, system (17) is split over the spatialvariables and the resulting one-dimensional equations are solved separately along each coordinate axis.However, the additional variations in the quantities at boundary points due to the fluxes in the correspondingdirections should be taken into account. This procedure is not required in PPM.

p2

Vi 1x------ V x( ) x.d

xi 1/2

xi 1/2+

=

VL

VR

Vi 1/2+ .= =

Ui 1/2+ j,U

LU

R+

2--------------------1

2--- *p

rUxp

V*( )p x*

p0>( )

1

2--- *p

rUxp

V*( ),p x*

p0


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Additional variations in the quantities, for example, at point 1between the cells (i,j) and (i+ 1,j) (seeFig. 9) are caused by the flux in they direction in the transition to the next time level. These variations canbe taken into account by solving Eq. (17) in thexdirection with the term responsible for the y directiontreated as a source. Then (24) is replaced by

(29)

(30)

where lpis a left vector of the Jacobian AandBpis a column of Bcorresponding to the ydirection. Thecomponents of lpandBpin (30) are calculated from the same state in which we fix the eigenvector basis inthe cell (i,j) for waves propagating along characteristics with positive eigenvalues or in the cell (i,j+ 1) forwaves corresponding to negative eigenvalues. The partial derivatives yVpare calculated by replacing themwith difference ones:

Here, and are taken from the previous time level. The solution to Eq. (29) is similar to (25)

but involves an additional source term:

(31)

The left VLand right VRboundary states at point 1are calculated by formula (23) by summing up the cor-responding amplitudes (31) times the right eigenvectors ofA. In what follows, the boundary state Vi+ 1/2,jiscalculated using formula (27).

The state Vi,j+ 1/2at point 2is calculated in a similar fashion. In this case, the source takes into accountthe variations caused by the flux in thexdirection.

6.2. Computation of the Fluxes on the Cell Boundaries

Figure 10 shows the set of characteristics corresponding to p> 0 inside the cell indexed by i.The char-acteristicx1(t) corresponds to the maximum eigenvalue 1 . Point 1lies at the intersection of this character-istic with the piecewise parabola at the previous time step. Obviously, the state at point 2(left boundary

state) is affected only by the domain between the cell boundaryx=xi+ 1/2and point 1.

To derive a second-order-accurate approximation in time, the amplitudes p(x, t)corresponding to eachcharacteristic have to be averaged over its domain of influence. For a wave propagating inside the cell i alongthe characteristic with p> 0, its averaged amplitude on the boundary p(x, t) at the time t+ is calculatedby the formula

(32)

tp px

p+ D

p, p 1 4,, ,= =

D

p

l

p

B

p

yVp

,=

yVp Vi j 1/2+,

pVi j 1/2,

p

y----------------------------------------- .=

Vi j 1/2,p

Vi j 1/2+,p

p xi 1/2+ t +,( ) p xi 1/2+ p t,( ) Dp.+=

i 1/2+p 1

p-------- p x( ) x, pd

xi 1/2+ p

xi 1/2+

0.>=

Vi i+ 1

Vi+1

Vi

VL

VR

xxi 1/2 xi + 1/2 xi + 3/2

Fig. 8.


13/20

1982


POPOV, USTYUGOV

The amplitudes p(x) can be calculated by multiplying (23) by the left eigenvectors:

(33)

Here, V(x)is taken from the previous time step. Relations (33) are substituted into (32) and the eigenvectors

lpfixed in each cell are taken outside the integral sign to obtain

where

(34)

We need to choose the state used to calculate the eigenvectors fixed in each cell. This state for the cell i

is defined as the contribution (averaged by formula (34)) to the left boundary state made by the wave

propagating along the characteristic corresponding to the maximum eigenvalue. Numerical computationshave shown that the numerical results are hardly affected by this choice.

Obviously, the waves propagating along the characteristics with p< 0 inside the cell ido not influencethe state at point 2. Therefore, any number can be taken for with p< 0. For convenience, we use thegeneral formula

(35)

The state to the left of the boundary is obtained via formula (23) by summing up amplitudes (35)

times the right vectors. However, it is more convenient to expand the increment of the state vector in termsof the right eigenvectors. To derive the expansion formula, (35) is written in the form

(36)

(here, the arguments of the eigenvectors were dropped.) Multiplying (36) by rpand summing the result overp, after simple rearrangements, we obtain

(37)

p x( ) lpV x( ), p 0.>=

i 1/2+p

lp

Vi 1/2+L p,

,=

Vi 1/2+L p, 1

p-------- V x( ) x, pd

xi 1/2+ p

xi 1/2+

0.>=

Vi 1/2+L 1,

i 1/2+p

i 1/2+p l

pVi 1/2+

L 1,( ) Vi 1/2+

L p,, p 0,>

lp

Vi 1/2+L 1,

( ) Vi 1/2+L 1,

, p 0.( )

=

1

2

y

x

i i + 1

j + 1

j

Fig. 9.


14/20



The cell ihas been considered thus far. A similar formula for determining to the right of the

boundary can be derived for the cell i+ 1 and a characteristic with negative eigenvalues:

(38)

Here, rpand lpare the eigenvectors fixed in the cell i+ 1. The state is the contribution to made

by the wave propagating along the characteristic corresponding to the maximum (in modulus) negativeeigenvalue averaged over the corresponding domain of influence in the cell i + 1.

Formulas (37) and (38) have an illustrative interpretation: they show that the increment in each physicalvariable near the boundary is the sum of its corresponding increments as each characteristic is crossed fromleft to right from one state to another.

To determine the flux at the boundaryx=xi+ 1/2, we again use Roes method (see [8]). First, we

proceed from the physical variables V to the conservative ones U(there is a one-to-one correspondence

V U) and compute the fluxes to the left and right of the boundary: FL= F( ) and FR= F( ).

Then we use the formula

(39)

where

(40)

and =Mrpis the right eigenvector for the conservative gas dynamics system (Mis the transition matrix

given by (18)). In (40), we used the relation

Due to this property, the left eigenvectors do not need to be computed in terms of conservative variables.

The components of V*are calculated using (28) with respect to the states and , and U*is the

state corresponding to V*in conservative variables.

In the two-dimensional case formulas (37) and (38) contain additional term (30). After the fluxes on thecell boundaries in thexdirection have been calculated, the fluxes in theydirection are determined in a sim-

ilar manner. Next, difference scheme (16) is used to compute the states at the cell centers at the next

time step.

Vi 1/2+R

Vi 1/2+R

Vi 1/2+R 1,

rp

lp

Vi 1/2+R p,

Vi 1/2+R 1,

( )( ).p p 0


15/20

1984


POPOV, USTYUGOV

To complete the solution of the problem, the boundary values and calculated by for-

mula (27) are updated according to the PPM method in the area of a nonmonotone solution (see (4), (5)).

7. TEST PROBLEMS IN GAS DYNAMICS

The method proposed was tested by solving several one- and two-dimensional gas dynamics problems.In all the cases, we considered the ideal-gas equation of state (15) with = 1.4.

7.1. One-Dimensional Tests

The first three tests in one dimension were performed according to [4]. Specifically, we considered Sodsproblem [9], Laxs problem [10], and Shus problem [11]. They were solved on the interval x [1, 1],which was initially divided into two subintervals by the boundaryxb:

We used historical boundary conditions; i.e., the boundary values did not vary with time.

The fourth test was the interaction of two shock waves (see [12]). This problem was solved on the inter-valx [0, 1], which was initially divided into three subintervals:

Total reflection conditions were used at the boundary in this test. For this purpose, we introduced three addi-tional cells with states V0, V1, and V2for the left boundary and three cells with states VN+ 1, VN+ 2, andVN+ 3for the right boundary. The values of these states at every time step were defined as

The CPU time was denoted by T, andNwas the number of difference cells. The time step was deter-mined by the Courant condition with the Courant number = 0.5 in all the computations.Test 1(Sods problem [9]). The initial conditions for the gas dynamics equations have the form

Test 2(Laxs problem [10]). The initial conditions have the form

Test 3(Shus problem [11]). The initial conditions have the form

Vi j, 1/2+n 1+

Vi 1/2+ j,n 1+

VV

L, x xb,

VR

, x xb.>

=

V

VL

, x 0.1, >0.5065 0.8939 0 0.35, , ,( ), x 0.5, y< 0.5,>1.1 0.8939 0.8939 1.1, , ,( ), x 0.5, y 0.5,< >2 0.75 0.5 1, , ,( ), x 0.5, y< 0.5,>1 0.75 0.5 1, , ,( ), x 0.5, y 0.5,

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