A numerical scheme for morphological bed level - James T. Kirby

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(2008) 167–180www.elsevier.com/locate/coastaleng

Coastal Engineering 55

A numerical scheme for morphological bed level calculations

Wen Long a,⁎, James T. Kirby b, Zhiyu Shao c

a University of Maryland Center for Environmental Science Cambridge, MD 21613, USAb Center for Applied Coastal Research, University of Delaware Newark, DE 19716, USA

c Department of Civil Engineering, University of Kentucky Lexington, KY 40506-0281, USA

Received 26 July 2007; accepted 5 September 2007Available online 15 October 2007

Abstract

The typical equation for bed level change in sediment transport in river, estuary and near shore systems is based on conservation of sedimentmass. It is generally a nonlinear conservation equation for bed level. The physics here are similar to shallow water wave equations and gasdynamics equation which will develop shock waves in many circumstances. Many state-of-art morphological models use classical lower orderLax–Wendroff or modified Lax–Wendroff schemes for morphology which are not very stable for long time sediment transport processessimulation. Filtering or artificial diffusion are often added to achieve stability. In this paper, several shock capturing schemes are discussed forsimulating bed level change with different accuracy and stability behaviors. The conclusion is in favor of a fifth order Euler-WENO scheme whichis introduced to sediment transport simulations here over other schemes. The Euler-WENO scheme is shown to have significant advantages overschemes with artificial viscosity and filtering processes, hence is highly recommended especially for phase-resolving sediment transport models.© 2007 Elsevier B.V. All rights reserved.

1. Introduction

The central concern of morphodynamics is to determine theevolution of bed levels for hydrodynamic systems such asrivers, estuaries, inlets, bays and other nearshore regions wherefluid flows interact with, and induce significant changes to, bedgeometry. The goal of this study is to describe a robustalgorithm for computing bed level change which is flexibleenough to handle the nonlinearity present in the equationsdescribing bed evolution in a range of physical settings.

Numerical morphological models involve coupling betweena hydrodynamic model, which provides a description of theflow field leading to a specification of local sediment transportrates, and an equation for bed level change which expresses theconservative balance of sediment volume and its continualredistribution with time. Recent examples of the development ofsuch models include the work of de Vriend et al. (1993),Nicholson et al. (1997), Kobayashi and Johnson (2001) andHudson et al. (2005). Because the flow field and the resulting

⁎ Corresponding author. 2020 Horns Point Road, Cambridge, MD 21613,USA.

E-mail address: [email protected] (W. Long).

0378-3839/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.coastaleng.2007.09.009

transport rate is a nonlinear function of bed level, the sedimentconservation equation is physically a nonlinear conservationequation for the bed level. The same situation occurs in otherphysics contexts, such as mass conservation equation inhydraulics and aerodynamics as well as traffic flow in highwaysystems (Whitham, 1974). A common feature of theseconservation laws is that shock waves, i.e. discontinuities ofthe respective physical quantities, will develop when particlevelocity approaches celerity. Several decades of research efforthas been devoted to the development of numerical solutiontechniques for obtaining accurate and stable simulations ofshock behavior.

As reviewed in Nicholson et al. (1997), many state-of-artmorphodynamic models use classical shock capturing schemesfor bed level simulation. For example, Johnson and Zyserman(2002) apply a second-order accurate modified Lax–Wendroffscheme (Abbott, 1978). The Delft Hydraulics model Delft2D-MOR (Roelvink and van Banning, 1994; Roelvink et al., 1994)uses a FTCS (forward time, central space) explicit scheme withcorrections of the transport rate to compensate negativenumerical diffusion resulting from the scheme. The HRWallingford model PISCES (Chesher et al., 1993) uses a one-step Lax–Wendroff scheme. STC (Service Technique Centraldes Ports Maritimes et des Voies Navigables) model uses a two-

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http://dx.doi.org/10.1016/j.coastaleng.2007.09.009

168 W. Long et al. / Coastal Engineering 55 (2008) 167–180

step Lax–Wendroff scheme (Tanguy et al., 1993). The Uni-versity of Liverpool model (O’Connor and Nicholson, 1989,1995) also uses a modified Lax–Wendroff scheme with effectsof gravity on the sediment transport rate. The Lax–Wendroffscheme suffers from dispersion resulting in spurious oscillationsoccurring in the numerical results; see, for example, Hudson et al.(2005). Various techniques, including flux-limiter methods, havebeen used to try to eliminate the spurious oscillations.Unfortunately, the spurious oscillations could not be eliminatedand overpowered the numerical results for long computational runtimes as also pointed out by Damgaard and Chesher (1997) andDamgaard (1998).

More recently, work has been focused on wave phase-resolving sediment transport models for nearshore applicationswhere waves play an important role as driving force (Long andKirby, 2003; Long et al., 2004; Hsu and Hanes, 2004;Henderson et al., 2004). The wave orbital velocities areoscillatory in time and space, and the resulting sedimenttransport fluxes are also oscillatory. It is difficult to obtainnumerical estimate of local bedform phase speed under thesecircumstances (see Appendix A). This weakens morphologicalschemes (such as Lax–Wendroff type of schemes) that requiredecent estimate of the phase speed. These complications makethe demand for better schemes more urgent in bed levelprediction.

In Hudson et al. (2005), a variety of numerical schemes arediscussed including versions of the Lax–Friedrichs (Lax, 1954)scheme, the classical Lax–Wendroff scheme, the MacCormack(1969) scheme and the Roe (1986) scheme based on shallowwater equation for hydrodynamics and simple power law forsediment transport rate. A flux-limited version of the Roescheme is found to be much more stable than Lax–Wendroffand Lax–Friedrichs type schemes. The disadvantage is that theRoe scheme involves calculation of eigenvectors for so-calledRoe averaged Jacobian matrix of the entire hydrodynamics andmorphology system. This is feasible for shallow water systemand simple power law sediment transport rates for 1-Dproblems. The numerics become tedious and complex for acoupled system of more comprehensive hydrodynamic andsediment transport models. 2-DH implementation of the Roescheme will be difficult in practice.

In this study, we investigate the stability and performance ofseveral finite difference schemes. We study the evolution (inone horizontal direction) of an initial mound of sediment in ananalytically tractable case with a rigid lid, in order to comparethe performance of various schemes. Two schemes based on theWeighted Essentially Non-Oscillatory (WENO) formalism ofLiu et al. (1994) are found to provide reasonably accuratereproduction of a simple shock structure in one horizontaldimension. Then, an Euler-WENO scheme, based on first orderexplicit time stepping together with a WENO scheme for spatialdiscretization, is applied in two examples based on depthintegrated, free surface flows. In the first example, a phaseresolving sediment transport model is used to study evolution ofperiodic sand bars in the presence of waves at the resonantBragg frequency. Finally, the evolution of periodic alternatingbars in an otherwise rectangular channel is considered.

2. Model equation for bed elevation in sediment transport

Bed level changes are governed by the equation for conser-vation of sediment mass, which can be written as

A

At1� np� �

zb þZ g

zb

cdz

� �þj � q ¼ 0 ð1Þ

where zb(x, t) is the bed level elevation (defined positive upwardrelative to a fixed datum) at each horizontal position x=(x, y) andtime t, np is the bed porosity, q=(qx, qy) is the total volumetricsediment transport rate, c(x, z, t) is the volumetric concentration ofsuspended sediments in the water column, η(x, t) is the freesurface elevation. The quantity (1−np)zb+ ∫zb

ηcdz represents thetotal sediment volume per unit horizontal area, either stored in thebed or suspended in thewater column. In the following, we neglecta detailed examination of the suspended load contribution, sincethe numerical instabilities to be considered are apparent in thereduced flux equation given by

AzbAt

¼ � 11� np

j � q ð2Þ

The transport rate q is a complex function of various hydrody-namic quantities such as currents, waves and water depth as wellas quantities associatedwith sediment properties such as sedimentdensity and grain size. No uniformly valid formulation forq existsat this time. Some experimental data and theory suggests that q isclosely related to some power of near bed fluid velocity u (Grass,1981; van Rijn, 1984a,b,c, 1993). Only under the simplestassumptions can q be written as a function of zb. Therefore, it isdesirable to have a morphology updating scheme that does notdepend on a particular form of the transport formula.

In this paper, we consider the application of several new shockcapturing techniques to the calculation of bedform evolution inunidirectional streams or oscillatory wave dominated flows.Several one-dimensional examples are considered in order toevaluate the schemes and illustrate their robustness, after which atwo-dimensional example of alternating bar instability in aunidirectional flow is considered. The schemes considered hereare developed using finite differencing on a centered grid structure.

3. Shock capturing schemes

There are numerous finite difference schemes for spatialdiscretization of advection-dominated equations. They can bedivided into two broad categories; central difference schemesand upwind schemes. Each has its own advantages anddisadvantages. Here, we present several schemes that havebeen used in numerical simulation of hydrodynamics andaerodynamics for shock capturing, and then introduce a schemebased on the Weighted Essentially Non-Oscillatory (WENO)framework. As far as discretization in time, many techniqueshave been utilized in the literature, such as simple Eulerdiscretization, multi-level schemes, predictor-corrector schemesand so on. We mainly focus on one and two step methods intime for sediment transport problems.

169W. Long et al. / Coastal Engineering 55 (2008) 167–180

3.1. Central difference schemes

The simplest central difference scheme is the FTCS (forwardtime, central space) scheme. The scheme is absolutely unstablefor hyperbolic conservation equations, but is conditionallystable for convection diffusion problems. The usual remedy toFTCS scheme for Eq. (2) is to introduce viscous effects. Laxand Wendroff (1960) scheme and Lax family schemes aremostly adopted by practitioners of computational fluiddynamics. Here, we compare the Lax–Wendroff scheme, theRichtmyer (1962) scheme and the MacCormack (1969) scheme.

The basic Lax–Wendroff scheme adopted in Johnson andZyserman (2002) and Kobayashi and Johnson (2001) is given by

znþ1bi

� znbi▵t

þ 1

1� np

qniþ1 � qni�1

� �2▵x

¼ ▵tC2i

2

znbiþ1� 2znbi þ znbi�1

� �▵x2

ð3Þwhere the left hand side is a simple FTCS scheme, and righthand side is an additional diffusion term which damps spuriousoscillations caused by the FTCS scheme. Its accuracy is secondorder in time and space. The stability condition of Eq. (3) isjCi

▵t▵x jV1 where Ci is a bedform propagation phase speed as

described in the Appendix. Use of this scheme requires anestimate of phase speed Ci, which is not very easy to estimate inmany instances (see Appendix). Overestimation of Ci willoverly smooth the solution, while underestimation of Ci makesthe scheme unstable.

In order to avoid calculating Ci, various two-step Laxschemes have been proposed. Here we examine the Richtmyer(1962) scheme and the MacCormack (1969) scheme. TheRichtmyer scheme is written as

znþ1=2bi

¼ znbiþ1þ znbi2

� ▵t2▵x

1

1� np� � qniþ1 � qni

� � ð4Þ

znþ1bi ¼ znbi �

▵t▵x

1

1� np� � qnþ1=2

i � qnþ1=2i�1

� �ð5Þ

and MacCormack scheme is written as

zbi ¼ znbi �▵t

▵x 1� np� � qni � qni�1

� � ð6Þ

znþ1bi ¼ znbi þ zbi

2� ▵t

2▵x 1� np� � q zbð Þiþ1�q zbð Þi

� � ð7Þ

The stability requirement of the Richtmyer scheme and theMacCormack scheme is jCi

▵t▵x jV1, and accuracy is second order

in time and space. Both schemes require calculation of sedimenttransport rate q at intermediate time levels corresponding to bedlevel zb

n + 1/2 and zb. This in turn can lead to costly recalculation ofhydrodynamics. This need is avoided in the Gaussian humpexample (see Section 5), where q can be specified completely as afunction of zb.

The common disadvantage of the second order centraldifference schemes is that strong spurious oscillations can begenerated near shocks or steep fronts. Generally, filtering and

artificial viscosity have to be added to make the solutions moreregular, or smooth, as illustrated below. In Johnson and Zyserman(2002), a filtering process suggested by Jensen et al. (1999) isused for the Lax–Wendroff scheme. The disadvantage of usingfiltering processes is that one may have difficulty deciding thenumber of filtering processes to be applied in practice.

3.2. Upwind schemes

Central difference schemes have higher order accuracy butmore restrictive stability requirements, and tend to generatespurious oscillations. Upwind schemes on the other hand aregenerally more stable due to inherent dissipation effects at theprice of lower order accuracy (for lower order upwind schemes).Here we choose the simplest first order upwind scheme FTBS(forward in time, backward in space) and the second orderWarming and Beam (1975) scheme.

The FTBS scheme is written as

znþ1bi

¼ znbi �▵t

2▵x 1� np� � 1� að Þ qniþ1 � qni

� �þ 1þ að Þ qni � qni�1

� ��

ð8Þ

with α=sign(Ci) This scheme is first order accurate in time andspace with strong dissipation. The scheme is conditionallystable for jCi

▵t▵x jV1, but has the disadvantage of widening the

shock region excessively.The Warming–Beam scheme adds a second order correction

to the FTBS scheme, which gives higher order accuracy but stillupwinding to achieve conditional stability. A first order (intime) version of the scheme can be expressed as

znþ1bi ¼ znbi �

▵t

1� np� �

▵xqiþ1=2 �qi�1=2

� �ð9Þ

where

qiþ1=2 ¼12

3qni � qni�1

� �� 12Ci�1=2 qni � qni�1

� �Ciþ1=2z 0

ð10Þ

qiþ1=2 ¼12

3qniþ1 � qniþ2

� �� 12Ciþ3=2 qniþ2 � qniþ1

� �Ciþ1=2b 0

ð11Þand

qi�1=2 ¼12

3qni�1 � qni�2

� �� 12Ci�3=2 qni�1 � qni�2

� �Ci�1=2z 0

ð12Þ

qi�1=2 ¼12

3qni � qniþ1

� �� 12Ciþ1=2 qniþ1 � qni

� �Ci�1=2b0 ð13Þ

with Ci+1/2, Ci− 1/2, Ci+3/2 and Ci− 3/2 the bed level phasespeed at respective locations. Again the second order correctionrequires knowledge of C which sometimes causes difficulty.


3.3. WENO Schemes

WENO (Weighted Essentially Non-Oscillatory) schemes arebased on the ENO (Essentially Non-Oscillatory) schemes ofHarten (1983) and Harten et al. (1987). The key idea of ENOscheme is to use the smoothed stencil among several candidates toapproximate flux q at cell boundaries (i±1/2) to high order and atthe same time to avoid spurious oscillations near shocks ordiscontinuities. WENO schemes take the process one step furtherby taking a weighted average of the candidate stencils. Weightsare adjusted to obtain local smoothness.More details can be foundin Liu et al. (1994), Jiang and Shu (1996), Jiang et al. (1999) andShao et al. (2004). Here we give a brief introduction for 1-Dproblem described by Eq. (2). WENO schemes have not beenused extensively for morphological calculations, and the resultshere represent the central contribution of this study.

The sediment transport rate q can be split into parts associatedwith bedform propagation in the positive and negative x direc-tions, given by q+ and q−.

qþ ¼ 1� np� � Z zb

0Cþ zð Þdz; ð14Þ

q� ¼ 1� np� � Z zb

0C� zð Þdz; ð15Þ

where C+=max(C, 0) and C−=min(C, 0). C+ and C− stand forphase speed of bedform propagating in positive x direction andnegative x direction respectively. Obviously, C=C+ +C−,hence q=q+ +q−. WENO scheme gives the following formulafor approximating term dq/dx:

dqdx

¼ qiþ1=2 � qi�1=2

▵xð16Þ

where the key is to estimate qi+1/2 and qi− 1/2, which areapproximations of the transport rate q at grid locations i+1/2and i−1/2. Again, qi+1/2 can be split into left-biased-flux q−i+ 1/2and right-biased-flux q +

i+1/2,

qiþ1=2 ¼ q�iþ1=2 þqþiþ1=2 ð17Þ

Here, left-biased-flux is calculated using

q�iþ1=2 ¼ x1q1iþ1=2 þ x2q

2iþ1=2 þ x3q

3iþ1=2;Ciþ1=2z 0 ð18Þ

q�1þ1=2 ¼ 0;Ciþ1=2b 0 ð19Þ

where

q1iþ1=2 ¼13qi�2 � 7

6qi�1 þ 11

6qi ð20Þ

q2iþ1=2 ¼ � 1

6qi�1 þ 5

6qi þ 1

3qiþ1 ð21Þ

q3iþ1=2 ¼13qi þ 5

6qiþ1 � 1

6qiþ2 ð22Þ

are three candidate stencils for estimating q at grid location i+1/2with third order accuracy (left-biased in the sense that 3 grid

points i−2 to i to the left of location i+1/2 are used but only 2grid points i+1 and i+2 to the right of location i+1/2 are used).ω1, ω2 and ω3 are carefully chosen weights such that q−i+1/2given by Eq. (18) is fifth order accurate approximate of q at gridlocation i+1/2 and near discontinuity no Gibbs phenomenaoccur. The calculation of weights is given by Jiang and Shu(1996) as

x1 ¼ a1a1 þ a2 þ a3

ð23Þ

x2 ¼ a2a1 þ a2 þ a3

ð24Þ

x3 ¼ a3a1 þ a2 þ a3

ð25Þ

where

a1 ¼ 0:1

S1 þ ϵð Þ2 ð26Þ

a2 ¼ 0:6


a3 ¼ 0:3


with ϵ≈10−20 is a small number to avoid division by zero and

S1 ¼ 1312

m1 � 2m2 þ m3ð Þ2þ 14

m1 � 4m2 þ 3m3ð Þ2 ð29Þ

S2 ¼ 1312

m2 � 2m3 þ m4ð Þ2þ 14

m2 � m4ð Þ2 ð30Þ

S3 ¼ 1312

m3 � 2m4 þ m5ð Þ2þ 14

3m3 � 4m4 þ m5ð Þ2 ð31Þ

Here, S1, S2 and S3 are called smoothness measurements and

m1 ¼ qi�2 ð32Þ

m2 ¼ qi�1 ð33Þ

m3 ¼ qi ð34Þ

m4 ¼ qiþ1 ð35Þ

m5 ¼ qiþ2 ð36ÞSimilarly, we can calculate right-biased-flux q+i+1/2 (3 grid

points to the right of grid location i+1/2 are used and only 2grid points to the left are used)

qþiþ1=2 ¼ x1 q1iþ1=2 þ x2 q

2iþ1=2 þ x3 q

3iþ1=2; Ciþ1=2b 0 ð37Þ

qþiþ1=2 ¼ 0;Ciþ1=2z0 ð38Þ


with

q1iþ1=2 ¼ � 1

6qi�1 þ 5

6qi þ 1

3qiþ1 ð39Þ

q2iþ1=2 ¼13qi þ 5

6qiþ1 � 1

6qiþ2 ð40Þ

q3iþ1=2 ¼116qiþ1 � 7

6qiþ2 þ 1

3qiþ3 ð41Þ

x1 ¼ a1a1 þ a2 þ a3

ð42Þ

x2 ¼ a2a1 þ a2 þ a3

ð43Þ

x3 ¼ a3a1 þ a2 þ a3

ð44Þ

a1 ¼ 0:3


a2 ¼ 0:6


a3 ¼ 0:1


S1 ¼ 1312

m1 � 2m3 þ m4ð Þ2þ 14

m2 � 4m3 þ 3m4ð Þ2 ð48Þ

S2 ¼ 1312

m3 � 2m4 þ m5ð Þ2þ 14

m3 � m5ð Þ2 ð49Þ

S3 ¼ 1312

m4 � 2m5 þ m6ð Þ2þ 14

3m4 � 4m5 þ m6ð Þ2 ð50Þ

and ν6=qi+3 in addition to equations following (32). Left-biased-flux q −

i−1/2 and right-biased-flux q+i−1/2 for location i−1/2 can be

calculated using the upper equations by simply shifting i backwardfor one step.

Finally, a simple Euler explicit scheme for temporaldiscretization can be used for updating zb; the resulting finitedifference scheme is referred to here as the Euler-WENOscheme and is given by

znþ1bi

� znbi▵t

þ 11� np

qiþ1=2 � qi�1=2

▵x¼ O ▵t;▵x5

� � ð51Þ

The Euler-WENO scheme still needs information about phasespeed Ci+1/2 and Ci−1/2, but it is less demanding than the Lax–Wendroff scheme or Warming–Beam scheme since only the signof Ci±1/2 is needed to judge the “upwind” direction. Because theWENO scheme is a nonlinear scheme in the sense that thecoefficients w1, w2, w3, ω1, ω2 and ω3 depend on the transportrate q adaptively rather than being constants, no theoreticalstability criterion is available. Time steps used in this studysatisfies the condition that Courant number is less than one.

3.4. TVD schemes

TVD (Total Variation Diminishing) schemes are designedsuch that the total variance of the solution TV ¼ Rþl

�l j AzbAx jdx

will remain constant or only decrease in time. During thesolution process, there will be no new extrema generated. Someclassical schemes satisfy TVD condition automatically, forinstance Lax–Friedrichs (Lax, 1954) scheme. Harten (1983)proposed a first order TVD scheme and a second order TVDscheme. Subsequently, many TVD schemes have beenproposed based on existing schemes. Delis and Skeels (1998)outline a number of schemes having the TVD property,including a Lax–Wendroff scheme (TVD-LW) and a MacCor-mack scheme (TVD-MC). Here, we follow Shu and Osher(1988) and Shao et al. (2004) to apply a TVD-Runge-Kutta(TVD-RK) scheme for third order time integration of Eq. (2).The TVD-RK scheme can be summarized as an algorithm of 5steps. The first step consists of an Euler forward step to get timelevel n+1.

znþ1b � znb▵t

þA 1

1�npq znb� �h i

Ax¼ 0 ð52Þ

The second step uses a second forward step to time level n+2.

znþ2b � znþ1

b

▵tþA 1

1�npq znþ1

b

� �h iAx

¼ 0 ð53Þ

The third step uses an averaging step to obtain an approxi-mation solution at n+1/2.

znþ1=2b ¼ 3

4znb þ

14znþ2b ð54Þ

The fourth step uses a third Euler step to get time level n+3/2.

znþ3=2b � znþ1=2

b

▵tþA 1

1�npq znþ1=2

b

� �h iAx

¼ 0 ð55Þ

Finally, the fifth step uses another averaging step finally toget solution at time level n+1,

znþ1b ¼ 1

3znb þ

23znþ3=2b ð56Þ

where the WENO scheme is used for the spatial discretizationThis scheme gives third order accuracy O(▵t3) in time and fifthorder accuracy O(▵x5) in space. But we also remark that in thesecond step and the fourth step, transport rate q has to berecalculated. It is relatively easy for the Gaussian hump test caseof this paper, but it can be costly for more calls to hydrodynamicmodules have to be made.

4. An initial hump in a unidirectional flow

In this section, we will apply the different schemes to a testcase involving an initially symmetric, isolated bedform, as inJohnson and Zyserman (2002) and Hudson et al. (2005).

Fig. 1. Simulation of evolution of Gaussian hump using the Lax–Wendroffscheme.


Assuming that transport rate q is a power function of currentspeed (Grass, 1981; van Rijn, 1984a,b,c, 1993), and assuming asteady discharge in a channel with a rigid lid, we have

q ¼ aub ð57Þu ¼ Q=h ð58Þh ¼ s� zb ð59Þwhere a, b are constants, h is the water depth, s is the datumwhich can be set to s=0, and Q is the constant fluid volume fluxper unit width. Following Johnson and Zyserman (2002),Eq. (2) can be written as

AzbAt

þ C zbð ÞAzbAx

¼ 0 ð60Þ

where C(zb) is the phase speed of bedform, expressed as

C zbð Þ ¼ 11� np

Aq

Azb¼ 1

1� np� �

s� zbð Þ abub ð61Þ

The following quantities are specified according to similarsettings in Hudson et al. (2005):

a ¼ 0:001 s2=m ð62Þb ¼ 3:0 ð63Þ

Q ¼ 10 m2=s ð64Þnp ¼ 0:4 ð65Þand the initial condition zb(x, 0) is given as a Gaussian hump,

zb x; 0ð Þ ¼ �h0 þ 2e �b x�xcð Þ2½ � ð66Þwith β=0.01 m−2, h0=6.0 m, 0≤x≤300 m and xc=150.0 m asthe center of the Gaussian hump. Eq. (60) may be solved by themethod of characteristics, which gives the result that zb willremain constant along characteristics given by dx/dt=C(zb).Further, since zb is constant along each characteristic, then dx/dtis constant and each characteristic is a straight line in {x, t} withslope given by C (zb(x, 0)) at its intersection with the x axis.Since, in the initial data dC/dxb0 for xNxc, characteristicsoriginating in this region will converge, leading eventually tomulti-valued solutions. In the following, we display weaksolutions which employ shock-fitting to develop a solutionsatisfying jump conditions, as indicated in Whitham (1974,Chapter 2). For corresponding numerical simulations, gridspacing is chosen to be ▵x=1 m, and the time step is chosen tobe ▵t=0.1 s, which is sufficiently small to satisfy stabilityconditions for all of the schemes discussed above.

Fig. 1 shows results for 10,000 s of evolution of the initialGaussian hump using the standard Lax–Wendroff schemewithout filtering. The scheme generates oscillations at the frontstarting around time t=1000 s, after which the initial crestbreaks into a row of dispersing, smaller isolated features. Thebasic features of this result, including the time to initialbreakdown of the profile into multiple crests and the number of

crests evolving out of the initial data, are common to the Lax–Wendroff, Richtmyer and MacCormack scheme results. Themultiple-crest behavior is indicative of frequency dispersioneffects in the partial differential equations corresponding to thedifference formulas, with the number of crests evolving in rank-ordered soliton-like behavior. Fig. 2 shows a comparison of thethree central difference scheme results to the analytical solutionto (56) at time t=600 s and t=2000 s. There are quantitativedifferences between the solutions but qualitative behavior isidentical across the family of schemes.

Fig. 3 shows the results of Lax–Wendroff scheme along withthe filtering process of Jensen et al. (1999) and Johnson andZyserman (2002), used once every 100 time steps. In the lowerpart of Fig. 3, bed level at time 0 s, 2000 s, 4000 s, 6000 s,8000 s and 10,000 s are plotted together against the analyticalsolution from the left to the right. We see that, compared toFig. 1, the filtered scheme propagates the solution withoutgenerating a dispersive train of bar features, but there areoscillations generated at the shock location and the width of thefront is not very well resolved.

Figs. 4 and 5 show results of the FTBS scheme and theWarming–Beam scheme. Both schemes propagate the barwithout generating multiple bar forms. The Warming–Beamscheme (Fig. 5) generates some oscillation at the bar front,including a small precursor dip in the solution. The FTBSscheme (Fig. 4) generates no oscillations but is quite dissipative(here implying a loss of sand; see below). The shock front isabout 5 grids wide, due to the strong damping effect of thescheme. Fig. 6 shows a comparison of filtered Lax–Wendroff,FTBS, Warming–Beam and analytical results at t=600 s, closeto analytic shock formation, and t=2000 s, after the shock hasformed. The dissipative nature of the FTBS scheme is clear.

Fig. 7 shows results for the Euler-WENO scheme; results forthe TVD-RK-WENO scheme are qualitatively similar asindicated in Fig. 8. Both schemes predict evolution of a smoothbar form with a sharply resolved shock front. The TVD-RK-WENO scheme is about 3 times slower than Euler-WENOscheme, without any significant quantitative change in results,


and hence we prefer the Euler-WENO scheme for practicalapplication.

All of the schemes tested here have been checked for volumeconservation. Expressing the total volume of sand as

V tð Þ ¼Z l

�lzb x; tð Þ � zblð Þdx; ð67Þ

where zb∞ is the bed level away from the hump. we compute arelative error E=(v(t)−v(0))/V(0) for each of the schemes. Allof the schemes are seen to be essentially conservative for the10,000 s of simulated time, with relative errors falling in therange of numerical roundoff.

Fig. 3. Simulation of evolution of Gaussian hump using the Lax–Wendroffscheme and filtering algorithm of Jensen et al. (1999); thin solid line — Lax–Wendroff scheme; thick solid line — analytical solution.

5. Application of the Euler-WENO Scheme to Sand BarDeformation due to Waves

Periodic rows of sandbars can strongly reflect surface waveswith twice the wavelength of the bar features through a processknown as Bragg scattering (Mei, 1985). Partial reflection ofwaves by a finite patch of bars creates a partial standing wave onthe upwave side of the bars. This perturbation to the incidentwave field has been shown to effectively promote the formationof new bars in an initially level sand bed (Rey et al., 1995).However, Yu and Mei (2000) have shown that the sedimenttransport mechanisms over an existing bar itself are destructive,leading to erosion of bar crests and infilling of bar troughs. As aresult, an evolving bar field would appear to march seawardover a flat bottom, with bars progressively growing on theseaward end and eroding at the landward end. This conclusionignores the possible influence of partial standing waves in theregion landward of the initial bar field, which would be apossible realistic complication if a reflective shore were present.

In this section, we apply the Euler-WENO scheme toinvestigate sand bar deformation under surface waves. Thecoupled evolution of bars and waves has been studiedtheoretically and experimentally by Yu and Mei (2000). When

Fig. 2. Comparison of central difference scheme results to analytical solution attimet=600 s (left) and t=2000 s (right). Solid line— analytic solution; ⁎— Lax–Wendroff; triangle — Richtmyer; circle — MacCormack.

wavelengths of bars and waves are comparable, the bars arecontrolled by a forced diffusion process due to partially standingwave groups and gravity. In Yu and Mei (2000), wave field issolved based on equations for incident and reflected waveamplitudes. The sediment transport rate is calculated byempirical bottom shear stress related to near bottom orbitalvelocity.

In the present study, we use the linearized Boussinesqequations to obtain the instantaneous wave field. The fluidparticle velocity adjacent to the bottom wave boundary layer isobtained using the quadratic vertical variation of horizontalflow assumed in Boussinesq wave theory. The bottom shearstress is obtained by solving for the turbulent wave boundarylayer with mixing length closure (Long et al., 2004). Diffusiondue to gravity is not included in this model, which will result inlarger growth rate of bars than seen in Yu and Mei (2000). Wealso point out that since the wave simulation, bed shear stresscalculation and sand transport rate calculation are different bothin formulation and parameters, the predicted bar morphology isnot deforming quantitatively at the same rate as Yu and Mei(2000).

Fig. 4. Simulation of Gaussian hump evolution using the FTBS scheme.

Fig. 5. Simulation of Gaussian hump evolution using theWarming–Beam scheme. Fig. 7. Simulation of Gaussian hump evolution using the Euler-WENO scheme.


Sediment transport is assumed to be dominated by bedloadcalculated using the Meyer-Peter and Müller (1948) formula

W x; tð Þ ¼ C1 h� hcð Þn ð68ÞwhereΨ is defined asW ¼ x; tð Þ= d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis� 1ð Þgdp� �

; q x; tð Þ is vol-umetric transport rate, d is sediment diameter, g is gravita-tional acceleration, s=ρs/ρ is specific gravity of sediments, θ=τb/(ρ(s−1)gd) is the Shields parameter, and θc is threshold value ofShields parameter for initiation of sediment transport,C1 and n aredetermined empirically.

The bed elevation is simulated using the Euler-WENOscheme with time stepping corresponding to that in thehydrodynamic model, and hence transport rate and bedelevation are being updated instantaneously to within theaccuracy of the time-stepping schemes. Initially, six sinusoidalsand bars are present at region 0bxb6π/k with bar amplitudeequal to near bottom excursion amplitude Ab, and wavelengthof bars equal to half the wavelength of waves. The sea bed isinitially flat elsewhere. Still water depth at the flat bed ish=7 m. Waves are introduced to the system from left boundaryx=−25/k with wave amplitude A0=0.5 m, wave period T=8 s

Fig. 6. Comparison of filtered Lax–Wendroff (circles), FTBS (stars), Warming–Beam (triangles) schemes with analytical solution (solid) at t=600 s (left) andt=2000 s (right).

and k being the incident wave number. Waves are absorbed atthe right boundary x=25/k such that reflected wave energy fromright boundary is nearly zero.

With the information given, we have Ab=A0 / sinh kh=0.64m.The time step of the simulation is chosen to be ▵t=0.05 s forwaves and morphology. The spatial grid size is ▵x=2 m. Otherparameters are ρs=2650 kg/m3, sediment diameter d=0.4 mm,still bed porosity np=0.3, C1=11.0, n=1.65.

Fig. 9 shows the surface elevation wave groups and the bedlevel at 4 chosen time spots. Fig. 10 shows the entire bed levelchange with respect to time.

According to Yu and Mei (2000), because initially each ofthe bar crest is upwave of a wave node by π/4 and downwave ofthe next antinode by π/4 (Fig. 9 a) and b)), the Bragg scatteredwaves deposit sands into the initial bar troughs and erode theinitial bar crests. Hence the bars will be flattened by theirscattered waves. New sand bars will grow offshore of the initialbars and their growth rate will be reduced as initial bars arediminishing. From Fig. 9, the above expectations are qualita-tively confirmed. The growing bars on the left have largergrowth rate due to the fact that waves are better sheltered on the

Fig. 8. Comparison of Euler-WENO scheme (stars) and TVD-RK-WENOscheme (triangles) results to analytical solution (solid line) at t=600 s (left),t=2000 s (middle) and t=6000 s (right).

Fig. 9. Bed level (b, f, d, h) and surface wave envelopes (a, c, e, g) at different times. The surface wave envelops are illustrated as a superposition of 19 snapshots of freesurface elevation 0.25 s apart from each other, where the central snapshot is taken at the labeled time.


right. Fig. 9 demonstrates that the Euler-WENO scheme isstable and fits in phase resolving sediment transport models.

6. Application of the Euler-WENO scheme to 2D channelstability problem

For 2D problems, the Euler-WENO scheme can be appliedsimply component-wise. In this section, we investigate the bedstabilities in a channel. The development of alternate bars instraight channels was investigated analytically by Colombiniet al. (1987) based on a linear and weakly nonlinear stabilityanalysis. It is shown that the initial perturbations of bed formwillgrow and reach an equilibrium amplitude, and the developmentof higher harmonics tends to cause diagonal fronts with highdownstream steepness. Schielen et al. (1993) show that thenonlinear evolution of the envelope amplitude of the group of

Fig. 10. Bed level change with respect to time.

marginally unstable alternate bars satisfies a Ginzburg–Landauequation and periodic bar pattern can become unstable,exhibiting quasi-periodic behavior for realistic physical para-meters and a dune covered channel bed. For increasinglyunstable bars, however, pronounced oblique shocks develop inthe alternating bar pattern (Chang et al., 1971). Güngördü andKirby (2001) studied the evolution of oblique shock structuresusing a pseudo-spectral scheme based on the hydrodynamicmodel of Özkan-Haller and Kirby (1997). The pseudo-spectralresults were similar to the results shown below in Fig. 11, butwere often contaminated by the presence of oscillations at thepoint of most rapid depth change in the down-channel direction,adjacent to channel side walls. The evolution of similar featuresin an initially constant depth sloping channel has also beenrecently studied by Defina (2003).

In this study, the channel geometry is defined as bounded bytwo straight walls at left (x=0) and right (x=Lx) respectively.The flow is driven by gravitational force in y direction.Sinusoidal initial channel bed level perturbations are imposed toinvestigate bedform instabilities. The average water depth is h0.The governing equations for the flow field are:

AhAt

þ APAx

þ AQAy

¼ 0 ð69Þ

APAt

þ AP2=hAx

þ APQ=hAy

þ ghAhAx

¼ �ghAzbAx

� fb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2 þ Q2ð Þp

P

8h2

ð70Þ

AQ

Atþ APQ=h

Axþ AQ2=h

Aþ gh

Ah

Ay¼ �gh

AzbAy

� fb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2 þ Q2ð Þp

Q

8h2þ i0gh

ð71Þ


where h(x, y, t) is total water depth, (P, Q) is depth integratedwater volume flux per unit width, i0 is the slope along thechannel, fb is bottom friction coefficient.

The transport rate formula used here is (Schielen et al., 1993;van Rijn, 1989)

q ¼ m0jujb ujuj � g0jzb

� �ð72Þ

where u=(P/h, Q/h) is the depth averaged velocity, ν0, b and γ′are empirical.

Fig. 11. Bed evolution of a straight channel (arrow: flow fiel

Periodic boundary conditions are employed in y direction(see Appendix). The computational domain is 0bxbLx and0bybLy. The initial bed level is given by

zb x; y; t ¼ 0ð Þ ¼ a0 cos k1 x cos k2y ð73Þ

Initial condition of the flow field is taken as still water (P=0,Q=0, h=h0− zb(x, y, t=0)).

The standard ADI (Alternating–Direction Implicit) schemeis implemented to solve the flow field, and the Euler-WENO

d u, averaged velocity about 1 m/s; contour: bed level).


scheme is used to update zb for the flow field calculation of thenext time step:

znþ1bi; j

� znbi; j▵t

þ 11� np

qxiþ1=2; j � qxi�1=2; j

▵x

þ 11� np

qyi; jþ1=2 � qyi; j�1=2

▵y¼ O ▵t;▵x5;▵y5

� �

ð74Þ

where (i, j) is grid location, qx and qy are x and y component of theWENO reconstructed sediment transport rate q at (i±1/2, j) and(i, j±1/2) respectively. The treatments of boundary conditions aredescribed in the Appendix.

Channel geometric parameters and flow parameters arechosen such that the flow is subcritical, and depth averagedvelocity |u| is about 1 m/s. The selected parameters are h0=5 m,i0=6.11×10

−5, fb=0.024, ν0=0.7, b=6, γ′=0.1, Lx=220 m,Ly=1310 m, k1=π/Lx, k2=2π/Ly, a0=1 m, np=0.3. Uniformspatial steps are used with ▵x=▵y=10 m. Time step of▵t=10 s is used.

The bed evolution in the channel is shown in Fig. 11. Theflow becomes fully developed rapidly. The bed level changes ina much slower time scale as expected. After 3 h, the initialsymmetric sinusoidal bed level has changed to be obliquelyoriented. The crests start to grow and pitch forward in the flowdirection, and the troughs become deeper and pitch against theflow direction. The distance between the contours indicates thegradient of the bed level is changing. These developmentsbecome more evident after 10 h. The bed form is also movingalong the flow direction. After 17 h, the bed form becomesstable, crests and troughs are distributed as alternating bars.

7. Euler-WENO scheme in generalizedcurvilinear coordinates

The present work was intended to provide the basis for amorphology calculation in several existing nearshore models.Several of these models, including the wave-averaged circula-tion model Shorecirc and the wave-resolving Boussinesq modelFunwave, have been extended to include grid geometries bsasedon generalized curvilinear coordinate systems (Shi et al., 2001,2003). The extension of the Euler-WENO scheme for thesediment conservation Eq. (2) is provided here.

The bottom elevation Eq. (2) can be written in curvilinearcoordinates as:

AzbAt

¼ � 11� np

j � q ¼ 11� np

1ffiffiffiffiffig0

p A

Axiffiffiffiffiffig0

pqi

� � ð75Þ

where qi is contravariant component of volumetric transportrate and

ffiffiffiffiffig0

pis the Jacobian of the generalized curvilinear

coordinateffiffiffiffiffig0

p ¼ xn1yn2 � xn2yn1 ; ð76Þwhere (x1, x2) = (ξ1, ξ2) are the curvilinear coordinates, and theyare transformed from Cartesian coordinates (y1, y2)= (x, y).

Since this equation bears the same form as Eq. (2) except thatthe differentiation is taken to

ffiffiffiffiffig0

pqi, the WENO reconstruction

of left-biased flux and right-biased-flux are kept the same asEqs. (18) and (37) except that q is replaced by

ffiffiffiffiffig0

pq to include

the effect of grid density change in curvilinear grid system.Separate WENO fluxes are constructed for sediment transport inξ1 direction contravariant component q1 and ξ2 directioncontravariant component q2 of total transport vector q=q1g1+q2g2=(q

1, qg). Here g1 and g2 are covariant basis vectors of thecurvilinear coordinates. The whole Euler-WENO scheme in 2Dcurvilinear coordinate (ξ1, ξ2) becomes

znþ1bi;j

� znbi;j▵t

þ 11� np

ffiffiffiffiffig0

pi; j

ðq1iþ1=2; j � q1i�1=2; jÞ▵n1

þ 11� np

ffiffiffiffiffig0

pi; j

ðq2i; jþ1=2 � q2i; j�1=2Þ▵n2

¼ O ▵t;▵n51;▵n52� �

ð77Þwhere qi + 1/2,j

1 and qi1− 1/2,j are WENO constructions of

ffiffiffiffiffig0

pq1 at

location (i+1/2, j) and (i−1/2, j) respectively; qi2+ 1/2,j and

qi2− 1/2,j are WENO constructions of

ffiffiffiffiffig0

pq2at location (i, j+1/2)

and (i, j−1/2) respectively. The construction process isidentical to Eqs. (18) and (37) except that here we need toinclude

ffiffiffiffiffig0

p.

8. Conclusions

A few selected classical and new shock capturing schemes arediscussed in the context of simulating bed level change due tosediment transport. Many are still using classical Lax–Wendroffand other central difference schemes which are of poorperformance and can not stand alone without filtering andartificial viscosity. Multi-time-level schemes are not suggestedfor bed level updating due to the cost of hydrodynamicscomputation. Upwind schemes are stable but have relatively lowaccuracy.

New shock capturing schemes such as WENO and TVDschemes are more appropriate for morphodynamics. Tests showthat the Euler-WENO scheme and the TVD-WENO schemewill be able to simulate bed level deformation with very goodaccuracy and stability in comparison to classical schemes. TheEuler-WENO scheme is preferred to the TVD-WENO schemebecause the TVD-WENO scheme requires much morecomputational cost.

It is also demonstrated briefly that the Euler-WENO schemeis successful in phase resolving sediment transport modelsunder waves. Investigation of alternating bar systems due tochannel instabilities shows the Euler-WENO scheme is alsocapable of modeling 2D morphology change.

A numerical scheme for realistic morphology calculationsneeds to be stable for extremely long integration times, and alsoneeds to be robust when used in conjunction with a variety ofstrategies for accelerated morphology updating. We have notaddressed these questions systematically here. Preliminarycalculations for the 1-D Gaussian bump case have beenperformed using the ’online’ formalism as defined by Roelvink


(2006) with morphology factors up to 100. For this simple case,the shock front structure is preserved accurately for this levelof accelerated updating. Much work remains to be done toevaluate this sort of application for the WENO scheme pro-posed here.

Acknowledgement

This work was supported by the National Ocean PartnershipProgram (N00014-99-1-1051) and the Office of NavalResearch, Coastal Geosciences Program (N00014-04-1-0219).

Appendix A. Numerical estimation of bedform phase speedC (zb)

For the 1-D case, Eqs.(2) and (6) give:

C zbð Þ ¼AqAx

1� np� �Azb

Ax

ð78Þ

For 2D case, the upper equation can be generalized(assuming that the bedform propagation direction is collinearas the gradient direction of zb) as:

C ¼ �AzbAt

jjzbj2jzb ¼ j � q

1� np� �jjzbj2

jzb; ð79Þ

where C=(Cx, Cy) is a vector. Eqs. (78) and (79) give thecalculations of phase speed in terms of the spatial derivatives.For the 1-D case, Hudson et al. (2005) employed a centraldifference scheme to estimate C numerically:

Ci ¼ qiþ1 � qi�1

1� np� �

zbiþ1 � zbi�1ð Þ ; zbiþ1p zbi�1: ð80Þ

Unfortunately, the upper equation can be used only whenzbi+1≠ zbi− 1 and it will produce quite inaccurate approximationof the phase speed when the bed level gradient approaches zeroor changes sign. For monotonic beach profile, this case doesn'toccur. For bedforms with sand bars, ripples or flat sea bed, it isprone to have significant errors.

In the WENO scheme implementation of this study, it isimportant that upwinding is used, and the easiest and the mostinexpensive way to achieve upwinding is to compute the Roespeed (Shu, 1997)

aiþ12¼ qiþ1 � qi

1� np� �

zbiþ1 � zbið Þ : ð81Þ

If ai+ 1/2≥0, the flow is from left to right, and correspondingbedform phase speed is also from left to right. Otherwise, ifai+1/2b0, the flow is from right to left. Since only the sign ofai+1/2 is needed, we can simply use sign((qi+ 1−qi)(zbi+ 1− zbi))to avoid division by a small number when (zbi+1− zbi) ap-proaches zero. Because WENO only requires the sign ofthe phase speed, it is much more stable than schemes thatrequire accurate estimate of phase speed both in magnitude andsign.

Appendix B. Boundary condition treatment

In Section 6, wall boundary conditions are applied at x=0and x=Lx, and periodic boundary conditions are applied at y=0and y=Ly. A staggered rectangular grid system is used for thewater flow simulation. At the wall boundaries x=0 and x=Lxthe boundary conditions of the flow are given as

Pjx¼0 ¼ 0 ð82Þ

Pjx¼Lx ¼ 0: ð83Þ

At the periodic boundaries y=0 and y=Ly, the boundaryconditions are

hjy¼0 ¼ hjy¼Ly ð84Þ

Qjy¼0 ¼ Qjy¼Ly : ð85Þ

In the ADI scheme that solves the flow Eqs. (69) to (71),boundary conditions (82) and (83) are realized by specifying zerovalues for P. Boundary condition (84) is satisfied indirectly bychoosing y=0 and y=Ly as nodes for fluxQ in the staggered gridsystem. Boundary condition (85) is realized by solving the cyclictridiagonal equation system (Sherman andMorrison, 1949)whichresults from the implicit discretization scheme in y direction forEq. (71). Note that one should only specify one of Eq. (84) orEq. (85) for a first order PDE system like Eqs. (69) (70) (71) andthis is also why here only P=0 conditions are applied in the xdirection.

The boundary conditions for sediment transport and morphol-ogy simulation are

qxjx¼0 ¼ 0 ð86Þ

qxjx¼Lx ¼ 0 ð87Þ

qyjy¼0 ¼ qyjy¼Ly ð88Þ

zbjy¼0 ¼ qyjy¼Ly ð89Þ

At the left most zb nodes zb1, j (index i=1 for zb and qx1, j islocated at (x=Δx/2, y)), Eq. (74) becomes

znþ1b1; j

� znb1; j▵t

þ 11� np

qx3=2; j � qx1=2; j▵x

þ 11� np

qyi; jþ1=2 � qyi; j�1=2

▵y¼ O ▵t;▵x5;▵y5

� �

ð90Þ

In this Eq. (90), we set qx1/2, j=0 according to (86) for qx1/2,j isthe interpolated sediment transport rate at (i, j) index (1/2, j), i.e.,(x, y)=(0, y). For the calculation of qx3/2, j≡ qx(1+1/2),.5j, accordingto Eqs. (17), (18) and (37), qx−1, j and qx 0, j are needed and theyare out of the domain as ghost points, which are filled out withvalues of −qx2, j and −qx1, j respectively to ensure qx is anti-symmetric (to the order ofO(▵x2)) about the boundary line x=0.Similarly, at the right most zb nodes zbM, j, we set qxM+1/2, j=0


according to (87) because index M+1/2 corresponds to x=Lx.Transport rates qx at the ghost points (M+1, j) and (M+2, j) arespecified in the same way as the left wall boundary.

In the y direction, at the bottom most nodes zbi,1, Eq. (74)becomes

znþ1b1; j

� znb1; j▵t

þ 11� np

qxiþ1=2; j � qxi�1=2; j

▵x

þ 11� np

qyi;3=2 � qyi;1=2▵y

¼ O ▵t;▵x5;▵y5� �

ð91Þ

where qyi,1/2 is interpolated sediment transport rate at (x, y)=(x,0). According to Eqs. (17), (18) and (37), in order to calculateqyi,1/2, qy at (i,−2), (i, −1), (i, 0), (i, 1), (i, 2) (i, 3) are needed. Herej=1, j=2, and j=3 are in the computational domain, yet j=−2,j=−1 and j=0 are ghost points. Because the problem is physicallyperiodic in the y direction, we set the values of qy at these ghostpoints to be qyi,N −2, qyi,N −1 and qyi,N accordingly. Similarly, atthe upper most nodes zbi,N, ghost points are at (i, N+1), (i, N+2)and (i, N+3) and the values of qy are set to be qyi,1, qyi,2, qyi,3accordingly. Since the transport rates are set to be periodic in the ydirection, the predicted bathymetry zb will be periodic in the ydirection.

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