Applied Mathematics and Computation 215 (2010) 3335–3342

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A composite semi-conservative scheme for hyperbolic conservation laws

Ritesh Kumar DubeyInstitut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse, France

a r t i c l e i n f o a b s t r a c t

Keywords:Hyperbolic conservation lawsNon-conservative schemeCentral and upwind difference methodsComposite schemes

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.10.022

E-mail address: rkdshru@gmail.com

In this work a first order accurate semi-conservative composite scheme is presented forhyperbolic conservation laws. The idea is to consider the non-conservative form of conser-vation law and utilize the explicit wave propagation direction to construct semi-conserva-tive upwind scheme. This method captures the shock waves exactly with less numericaldissipation but generates unphysical rarefaction shocks in case of expansion waves withsonic points. It shows less dissipative nature of constructed scheme. In order to overcomeit, we use the strategy of composite schemes. A very simple criteria based on wave speeddirection is given to decide the iterations. The proposed method is applied to a variety oftest problems and numerical results show accurate shock capturing and higher resolutionfor rarefaction fan.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Consider the 1D scalar conservation equation,

@u@tþ @f ðuÞ

@x¼ 0; ðx; tÞ 2 R� Rþ; ð1Þ

together with the initial condition

uðx; t ¼ 0Þ ¼ u0ðxÞ; ð2Þ

where u 2 R and the flux function f ðuÞ : R! R is a non-linear convex function of conservative variable u. Numerical studyand simulation of hyperbolic problems is non-trivial and of great importance as they often arise in the modeling of severalphysical phenomena in the areas such as acoustics, aero-acoustics, weather prediction and ground water flows etc. A gooddetail on various numerical schemes can be found in [1–5]. The numerical schemes developed for hyperbolic conservationlaws are mostly conservative as conservative form is mandatory for capturing the shock at right location for flows containingshock waves [8]. On the other hand non-conservative schemes works well for the smooth solution but these methods aresimply discarded by CFD community as they often converge to wrong week solution and captures the shock at wrong posi-tion. Recently it is reported that conservative methods produce anomalous solutions for special but important hyperbolicproblems for which conservative form is not known [4,19]. A class of conservative schemes known as high resolutionschemes (e.g. [7,10,17,6,12]) are much used by CFD community. In [18] some high-resolution conservative schemes (e.g.,[9,15]) are studied and are reported to work very inefficiently for special cases and give inaccurate shock position evenfor very fine mesh. In fact, it is also shown in [8] that if a stable non-conservative approximation is locally corrected, i.e.,in the vicinity of shock by a conservative scheme then such hybrid scheme converge to weak solution of non-linear conser-vation laws.

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3336 R.K. Dubey / Applied Mathematics and Computation 215 (2010) 3335–3342

These observations motivate us to look for a semi-conservative scheme which remains non-conservative for the region ofsmooth solution and locally changes into conservative form around discontinuities so that it can capture shock accurately.We aim to construct a scheme which is able to utilize the information on wave speed and also be able to capture the shockdiscontinuity at right location. In order to do so we use the non-conservative form of conservation law to have explicit infor-mation of the characteristic speed and construct an upwind scheme which respect physical hyperbolicity. The semi-conser-vative scheme captures the shock discontinuity correctly but in the presence of sonic point where wave speed changes itssign, produces entropy violating solution. In order to overcome this, we use the idea of composite schemes in which a base-line numerical method is coupled with another numerical method having complementary features as a filter [13,11]. Wepropose to use an entropy satisfying complementary scheme such as E-Schemes [16] or Lax–Friedrichs scheme in presenceof sonic point. We give a simple criteria based on wave speed for deciding such iteration.

2. Semi-conservative formulation

Consider the transport equation

@u@tþ a

@u@x¼ 0; a – 0: ð3Þ

We discretize the domain with fix space and time step size Dx; Dt, respectively, and let Ui ¼ Uni � uðxi;nDtÞ. It is well-known

that the first order upwind approximation for (3) is given by,

Unþ1i ¼

Ui � akðUi � Ui�1Þ; a P 0;Ui � akðUiþ1 � UiÞ; a 6 0;

�ð4Þ

where k ¼ DtDx. Re-write (1) in non-conservative form as,

@u@tþ aðuÞ @u

@x¼ 0; ð5Þ

where aðuÞ ¼ @f ðuÞ@u is the characteristic speed for (1). On comparison with (3) and utilizing the explicit information of the char-

acteristic speed we can construct a non-conservative upwind scheme so that it respect physical hyperbolicity property asso-ciated with (1). Also, since the conservative form is required in the vicinity of discontinuities, we propose the followingupwind discretization for (5):

Unþ1i ¼

Ui � DtDx ai�1

2ðUi � Ui�1Þ; if aiþ1

2P 0; ai�1

2P 0;

Ui � DtDx aiþ1

2ðUiþ1 � UiÞ; if aiþ1

26 0; ai�1

26 0:

(ð6Þ

Here ai�12

is the approximate characteristic speed of (1) and is given by,

aiþ12¼

Fiþ1�FiUiþ1�Ui

; if Uiþ1 – Ui;

f 0i ¼ f 0ðUiÞ; if Uiþ1 ¼ Ui;

(ð7Þ

where Fi ¼ f ðUiÞ.

3. Composite scheme

We observed that the proposed semi-conservative scheme captures the shock discontinuity correctly but in presence ofsonic point, produces entropy violating expansion shocks which shows the less dissipative nature of semi-conservativescheme near sonic point (see Fig. 3). We use the idea of composite scheme [13] to couple this semi-conservative scheme withanother numerical method having complementary features as a filter. Since the semi-conservative upwind scheme is lessdissipative, therefore, to overcome this, we propose to use a conservative diffusive Lax–Friedrichs (LxF) scheme as comple-mentary scheme.

Unþ1i ¼

Ui � DtDx ai�1

2ðUi � Ui�1Þ; if aiþ1

2P 0; ai�1

2P 0;

Ui � DtDx aiþ1

2ðUiþ1 � UiÞ; if aiþ1

2< 0; ai�1

2< 0;

ðUiþ1þUi�1Þ2 � Dt

2Dx ðf ðUiþ1Þ � f ðUi�1ÞÞ; else:

8>><>>: ð8Þ

4. Analysis

In this section we study the stability and conservative behavior of proposed composite scheme (8) in the vicinity ofdiscontinuities.

R.K. Dubey / Applied Mathematics and Computation 215 (2010) 3335–3342 3337

Definition 1. A one step numerical scheme given by [5,2]

Unþ1i ¼

Xk

ckUniþk; ð9Þ

is said to be monotonicity preserving if and only if

ck P 0; 8k:

Theorem 1. If the CFL like condition

C ¼ k maxi

aiþ12

��� ��� 6 1; k ¼ DtDx

; ð10Þ

is satisfied then the composite scheme (8) is monotonicity preserving.

Proof. Consider the composite scheme (8) with the following cases:

Case 1: aiþ12

P 0; ai�12

P 0; we have resulting scheme as

Unþ1i ¼ 1� kai�1

2

� �Ui þ kai�1

2Ui�1: ð11Þ

Case 2: aiþ12< 0; ai�1

2< 0; we have resulting scheme as,

Unþ1i ¼ 1þ kaiþ1

2

� �Ui � kaiþ1

2Uiþ1: ð12Þ

Obviously by definition, (11) and (12) will be monotonicity preserving, respectively, if,

kjaiþ12j 6 1; 8 i: ð13Þ

Case 3: For all other possible cases composite scheme approximate the solution of (1) by Lax–Friedrichs scheme which isconservative and monotonicity preserving under the CFL like condition (10) [2,4]. h

Now we show that in the vicinity of shock discontinuity, proposed scheme gives conservative approximation. Let X be theregion of discontinuity or non-smooth region of the solution and di ¼ fi� 1; i; iþ 1g be the neighbourhood of grid point i 2 Xs.t. di # X. Then it can be seen that 8i 2 X there exists atleast one grid point j 2 di s.t., Uj – Ui if j – i. Hence in such regions X,scheme (6) and the definition of characteristic speed aiþ1

2in (7) give,

Unþ1i ¼

Uni � Dt

Dx ðFi � Fi�1Þ; if aiþ12

P 0; ai�12

P 0;

Uni � Dt

Dx ðFiþ1 � FiÞ; if aiþ12< 0; ai�1

2< 0;

(ð14Þ

which can be written in conservative form for (1) as

Unþ1i ¼ Un

i � k Hiþ12� Hi�1

2

� �; ð15Þ

where

Hiþ12¼

Fi; aiþ12

P 0;

Fiþ1; aiþ12< 0:

(ð16Þ

4.1. Remarks

(1) Composite semi-conservative scheme is monotonically stable and is locally corrected by a conservative scheme which

ensures its convergence to weak solution of non-linear conservation laws [8].

(2) The proposed scheme is different from the classical conservative upwind scheme for (1). In fact, in smooth solutionregion the proposed scheme (6) may give non-conservative approximation to (1).

5. Numerical results

5.1. Convex flux function: inviscid burgers equation

We consider the inviscid Burgers equation given by,

@u@tþ @

@xu2

2

� �¼ 0; t > 0; ð17Þ

3338 R.K. Dubey / Applied Mathematics and Computation 215 (2010) 3335–3342

with periodic boundary conditions. It is well-known that the solutions of inviscid Burgers equation may contain shocks [1].We present three different test cases to show the accuracy and high resolution shock capturing ability of proposed compositescheme.

5.1.1. Case 1: accuracy test pre- and post-shockIn this example, we take smooth sinusoidal initial condition,

Table 1Solution

N

10

20

40

80

160

320

uðx; 0Þ ¼ �sinðxÞ; x 2 ½0; 2p�: ð18Þ

It is a consequence of non linearity of (17) that the (non-trivial) solution beginning from smooth initial conditions may de-velop a finite-time derivative singularity.

Table 1 demonstrates L1 and L1 norms of the errors of proposed composite scheme (C-scheme) and Lax–Friedrichs (LxF)scheme for C ¼ 0:4 at pre-shock time t ¼ 0:5 when the solution is still smooth. Table 1 shows error reduction due to doublingof the number of intervals is comparable with Lax–Friedrichs scheme. In fact, point wise errors by composite scheme aresignificantly smaller than the LxF scheme. It shows the first order accuracy of the proposed composite scheme. In Fig. 1,approximate solutions are presented at post-shock time t ¼ 2:0 when the shock is well developed. Graph shows that the pro-posed scheme captures the shock at right location with very high resolution even for N ¼ 21 grid points.

5.1.2. Case 2: shock and simple rarefaction fanHere we consider the Burgers Eq. (17) along with the following initial condition:

uðx;0Þ ¼1; for jxj < 1=3;0; for jxj > 1=3:

�ð19Þ

This test case has no sonic point in its solution. Initial jumps at x ¼ �1=3 and x ¼ 1=3 evolves into a simple rarefaction fanand a strong right shock, respectively. Computed results are given in Fig. 2 which show that the proposed scheme capturesthe left rarefaction fan and right shock at right location with higher accuracy.

5.1.3. Case 3: shock and rarefaction waves in presence of expansive sonic pointConsider the Burgers Eq. (17) with following initial condition:

uðx;0Þ ¼1; for jxj < 1=3;�1; for jxj > 1=3:

�ð20Þ

In this case, the jump at x ¼ �1=3 creates a simple centered rarefaction fan and the jump at x ¼ 1=3 creates a strong steadyshock. The unique sonic point is zero [1] and the wave speed f 0ðuÞ changes sign from �1 to 1. Computed solutions of pro-posed scheme (Scheme) and composite scheme (C-scheme) with Lax–Friedrichs method (LxF) are presented in Figs. 3 and4. The proposed semi-conservative scheme (Scheme) captures the steady shock at right location without any numerical dis-sipation but fails to resolve the left rarefaction waves and produces unphysical rarefaction shock Fig. 3. The results in Fig. 4show that the LxF captures shock wave and rarefaction fan but overly diffusive and produces step-wise solution profile. Theproposed composite scheme (C-scheme) approximates the rarefaction waves with higher resolution and captures the strongsteady shock at right location.

error behavior in terms of error ratios (E.R.) with the mesh refinement for C ¼ 0:4 and t ¼ 0:5.

Composite scheme LxF

L1 norm E.R. L1 norm E.R. L1 norm E.R. L1 norm E.R.

0.051910 – 0.169007 – 0.189060 – 0.394568 –

0.020618 2.51 0.049600 3.40 0.111464 1.69 0.235387 1.68

0.013920 1.48 0.038374 1.29 0.061314 1.82 0.137333 1.71

0.008171 1.70 0.036498 1.05 0.032406 1.89 0.075532 1.82

0.004443 1.84 0.022865 1.59 0.016910 1.91 0.040147 1.88

0.002345 1.89 0.012683 1.80 0.008652 1.95 0.020223 1.98

0.001260 1.86 0.006708 1.89 0.004278 2.02 0.009644 2.09

C=0.5

N=40

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

ExactC−scheme

LxF

ExactC−scheme

LxF

C=0.8

N=20

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

Fig. 1. Comparison of shock capturing property at time t ¼ 2:0.

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

LxFExact

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

SchemeExact

Fig. 2. Comparison of numerical results for C ¼ 0:5;Dx ¼ 0:02: at time t ¼ 0:6.

−1

−0.5

0

0.5

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

ExactScheme

Fig. 3. Exact shock capturing but entropy violating unphysical left rarefaction shock for data C ¼ 0:8;Dx ¼ 0:03 at time t ¼ 0:3.

R.K. Dubey / Applied Mathematics and Computation 215 (2010) 3335–3342 3339

5.2. Convex–concave flux: Buckley–Leverett equation

The case that flux function of conservation law is neither convex nor concave can be illustrated by the example of thescalar one-dimensional version of the Buckley–Leverett equation. This equation physically represents the flow of a mixtureof oil and water through a porous medium.

@u@tþ @f ðuÞ

@x¼ 0: ð21Þ

The flux function is given by,

f ðuÞ ¼ u2

u2 þ að1� uÞ2: ð22Þ

−1

−0.5

0

0.5

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Exact

LxF

Exact

−1

−0.5

0

0.5

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

C−scheme

Fig. 4. Comparision for the left rarefaction fan and right shock capturing property for data C ¼ 0:8;Dx ¼ 0:03 at time t ¼ 0:3.

3340 R.K. Dubey / Applied Mathematics and Computation 215 (2010) 3335–3342

Here a is viscosity ratio and u represents the saturation of water and lies between 0 and 1.

5.2.1. One moving shock [5]Consider Eq. (21) with a ¼ 1

2 and initial condition

0

0.2

0.4

0.6

0.8

1

−

uðx;0Þ ¼1; x < 0;0; x > 0:

�ð23Þ

The solution involves one single moving shock followed by an rarefaction wave. Numerical results using LxF and proposedsemi-conservative scheme (Scheme) are compared and presented in Fig. 5. Proposed scheme sharply captures the movingshock and rarefaction wave with higher resolution for the conners.

5.2.2. Two moving shock [14]Consider Eq. (21) with a ¼ 1

4 and subject to initial condition

uðx;0Þ ¼1; �0:5 6 x 6 0;0; elsewhere:

�ð24Þ

The solution involves two moving shocks, each followed by an rarefaction wave. Numerical results are given in Fig. 6 whichshow that the proposed scheme sharply captures both the fast and slow shocks. The rarefaction waves are also approximatedwith high resolution.

5.2.3. Flux sin problemThis scalar Riemann problem with non-convex flux function is introduced by Leveque [3]. It is given by,

@u@tþ @ðsinðuÞÞ

@x¼ 0; ð25Þ

with initial condition,

1 −0.5 0 0.5 1 1.5 2 2.5

ExactLxF

0

0.2

0.4

0.6

0.8

1

−1 −0.5 0 0.5 1 1.5 2 2.5

ExactScheme

Fig. 5. Solution profile for data C ¼ 0:5;Dx ¼ 0:02 at time t ¼ 1:0: composite scheme gives sharp resolution for rarefaction fan.

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

ExactLxF

0

0.2

0.4

0.6

0.8

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

ExactScheme

Fig. 6. Solution profile using C ¼ 0:4;Dx ¼ 0:01 at time t ¼ 0:4: composite scheme gives sharp resolution for both shocks and rarefaction fans.

0

2

4

6

8

10

12

14

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

ExactLxF

Scheme 0

2

4

6

8

10

12

14

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

ExactC−scheme

Fig. 7. Solution profile using C ¼ 0:2;Dx ¼ 0:02 at time t ¼ 1:0.

R.K. Dubey / Applied Mathematics and Computation 215 (2010) 3335–3342 3341

uðx;0Þ ¼p4 ; x < 0;15p

4 ; x > 0:

(ð26Þ

The solution consists one stationary shock at the origin, one moving shock followed by an rarefaction wave and one movingrarefaction wave. Solutions obtained by LxF and semi-conservative scheme are shown in Fig. 7(Left). LxF scheme shows itsstepwise behavior while the semi-conservative scheme (Scheme) captures the stationary shock exactly but fails to resolveboth the rarefaction waves. Proposed composite scheme (C-scheme) resolves both the rarefaction waves are captured withbetter resolution and captures the stationary shock and the moving shock with little dissipation.

6. Conclusion and future work

In this work, the non-conservative form of conservation law is used to construct a semi-conservative scheme which be-comes conservative in the vicinity of discontinuity. Proposed scheme captures the shock with high accuracy but yields rar-efaction shocks in presence of sonic point. A simple criteria is proposed to utilize the idea of composite scheme to avoid suchunphysical solution. The proposed composite scheme gives good result when applied to various convex, convex–concavescalar test problems. Extension for the high order accurate scheme is under investigation.

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