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FINITE DIFFERENCE SCHEMES FOR SCALAR CONSERVATION LAWS WITH SOURCE TERMS HANS JOACHIM SCHROLL



AND RAGNAR WINTHER



| IMA J. Num. Anal., 1996(16), pp 201-215. |

Abstract. Explicit and semi{implicit nite dierence schemes approximating nonhomogenous scalar conservation laws are analyzed. Optimal error bounds independent of the stiness of the underlying equation are presented. Key words. Hyperbolic conservation law, source term, nite dierence scheme, error estimate. AMS(MOS) subject classi cations. 35L65, 65M06, 65M15

1. Introduction. The purpose of the present paper is to study nite dierence

schemes applied to the Cauchy problem for nonhomogenous scalar conservation laws of the form:

ut + f (u)x = g(u):

(1.1)

From a numerical point of view, one obviously has to distinguish the sti and the nonsti case. According to Pember (1992), the conservation law (1.1) is sti if the time scale introduced by the source term g is small compared to the characteristic speed f 0 and some appropriate length scale. The generic form of a sti conservation law is (1.2) ut + f (u)x = 1 g(u); where the relaxation time  is a small parameter while the equation (1.1) corresponding to  = 1 is referred to as the nonsti case. The schemes that shall be analyzed are modi cations of Godunov's method. In case of the nonsti equation (1.1) we analyze the explicit method:

unj +1 ? unj f (u(unj ; unj+1)) ? f (u(unj?1; unj)) (1.3) = g(unj):
t +
x Here u(u?; u+) is the solution of the homogenous Riemann problem ( ? u ; x 0 the solution of the Riemann problem is given by the left state u(u?; u+) = u? and both schemes reduces to simple backward dierence methods. We are particularly interested in estimating the global error of both schemes in L1. In the nonsti p case we will prove, that if the proper CFL-condition is satis ed, the 1 L -error is O(
t). This corresponds to a well known result for homogenous, scalar conservation laws, see eg. Kuznetsov (1976) or Lucier (1985). It should be mentioned here that it is not dicult to extend Lucier's argument to the nonhomogenous case and to obtain an error estimate for the real Godunov scheme applied to (1.1) or (1.2). The problem is that in this situation Godunov's method does not reduce to a computable scheme, since it requires the exact solution of the conservation law with the source term included on small time intervals. In the p sti case we shall establish under proper additional assumptions on g that the O(
t) estimate holds independently of the relaxation time . For a explicit scheme similar to (1.3) convergence towards the entropy solution of (1.1) was proved by Chalabi (1992). As it is based on Helly's theorem, his result does not give any information on the rate of that convergence. A semi{implicit scheme of type (1.4) applied to a special sti system has been analyzed by Schroll, Tveito & Winther (1994). The argument presented in the present paper is strongly related to the technique introduced by Schroll et al. (1994). We will refer to that paper for several details. The present paper is organized as follows: In Section 2 we introduce the notation, give the exact assumptions on both models (1.1) and (1.2) and review properties of the entropy solution. Furthermore, the main results are stated. In Section 3 we prove the error estimate in the nonsti case, and Section 4 is devoted to the semi{implicit scheme and the sti case. Finally, Section 5 provides proofs for some auxiliary results. 2. Preliminaries and the main results. Throughout the paper, we assume the

ux f and the source g to be given smooth functions satisfying g(0) = 0: We shall consider solutions of the Cauchy problem with given initial data

u0 2 L1(R) \ BV (R; R): Here, BV denotes the subspace of L1loc consisting of functions with bounded total variation Z jz (x + h) ? z (x)j dx: TV (z) := sup jhj h6=0 R

2. PRELIMINARIES AND THE MAIN RESULTS

3

Given the above assumptions, it is well known that there is a unique entropy solution of the Cauchy problem for (1.1) or (1.2). This solution is de ned by the so{called Kruzkov inequality (Kruzkov 1970, de nition 1): Z TZ

0 R

Z

(2.1)

[ju ? kj't + (u ? k)(f (u) ? f (k))'x ] dx dt

+ R[ju0 ? kj'(x; 0) ? ju(x; T ) ? kj'(x; T )] dx Z TZ 1  ?  0 R[(u ? k)g(u)'] dx dt; for any T > 0; k 2 S and any ' 2 D+ (T ):

Here,  is the sign function and D+ (T ) is the set of all nonnegative C 1{functions with compact support in R  [0; T ]. Throughout this paper, the nal time T > 0 will be considered xed. The solution of the nonsti equation (1.1) is assumed to be located in the state space

S = fu 2 R : juj  g for a suitable positive constant . Furthermore, if

Lg = sup jg0(u)j u2S

we assume that (2.2)

exp(Lg T )ku0k1  ;

where k
k1 denotes the L1-norm. The inequality (2.2) will imply that the entropy solution u stays in the state space, i.e. u(x; t) 2 S for t 2 [0; T ]. Also, since g(0) = 0, (2.3)

jg(u)j  Lg juj; 8u 2 S :

The assumptions above also imply that there is a constant M , depending on TV (u0), such that (2.4)

TV (u(
; t))  M; 8t 2 [0; T ];

and (2.5)

ku(
; t) ? u(
;  )k1  M jt ?  j; 8t;  2 [0; T ]:

In fact, the TV {bound (2.4) and the L1{continuity (2.5) follow from Lemma 3.1 below. Next, we describe precisely the schemes which shall be analyzed. Choosing positive grid sizes
t and
x in t and x direction, we de ne cell boundaries xj+1=2 := (j +1=2)
x

2. PRELIMINARIES AND THE MAIN RESULTS

4

and corresponding computational cells Ij := [xj?1=2; xj+1=2). The initial data for the scheme is computed as cell averages of u0 Z u0j =
1x I u0(x) dx: (2.6) The discrete solution funjg generated by either of the schemes (1.3) or (1.4) is considered as approximation of the average of the entropy solution in Ij at time level tn := n
t. Due to the explicit treatment of the convective terms, it is reasonable to assume a CFL{condition
t jf 0(u)j  1;  :=
(2.7) x for all relevant values of u. Now, we are in the position to state the result in the nonsti case: Theorem 2.1. Assume that the conditions given above are satis ed. Let u be the entropy solution of (1.1) and funjg the approximations generated by (1.3). Then there is a constant M ,independent of
t and
x, such that j

(2.8)

p ku(
; tn) ? un k1  M
t; 0  tn  T;

where un denotes the piecewise constant function representing unj . For the sti equation (1.2) we would like to have a similar result, where the estimate (2.8) holds independently of the relaxation time . Several other authors observed that such a result can not hold for general source terms, i.e. source terms that allow more than one equilibrium state (LeVeque & Yee 1990, Berkenbosch, Kaasschieter & ten Thije Boonkkamp 1994, Klingenstein 1994). Therefore, we need additional assumptions to deal with the sti equation. The assumption we propose is that the source g decreases strictly with respect to u. Therefore, zero is the only equilibrium. More precisely, the function g is assumed to satisfy

(2.9)

g(0) = 0;

g0(u)  ? < 0 8u 2 [0; 1]:

If g satis es this property, then the unit interval is a time invariant state space for the equation (1.2). Furthermore, it will be assumed below that the initial data is "close to equilibrium". We emphasize that the only stability condition that is assumed is the CFL{condition (2.7) with respect to the unit interval. Theorem 2.2. In addition to the assumptions given in Theorem 2.1 assume that g satis es (2.9). Furthermore, the initial data satis es u0 2 BV (R; [0; 1]) and there is a constant M0 independent of  > 0, such that kg (u0)k1  M0. Then the semi{implicit scheme (1.4) has a unique solution unj 2 [0; 1]. Furthermore, if u is the entropy solution of (1.2) there is a constant M , depending on M0, but independent of
t;
x and , such that (2.10)

p ku(
; tn) ? un k1  M
t; 0  tn  T:

3. THE NONSTIFF CASE

5

Here, we want to point out that due to the implicit treatment of the source term p the error bound does not depend on . Furthermore, both error estimates of order O(
t) are optimal, since the same estimate is optimal for homogenous conservation laws, cf. Lucier (1985). The outline of the arguments for both proofs is as follows: In order to compare the approximation generated by the scheme to the entropy solution, we de ne a comparison function u
on R  R+ which interpolates the data unj in the sense that (2.11)

u
(y; t+n ) = lim u (y;  ) = unj for y 2 Ij : !t
n  >tn

This comparison function will be discontinuous at the discrete time levels tn. Furthermore, a Kruzkov{type inequality for u
will be derived which can be seen as a weak entropy formulation of the scheme. Using this formulation and the original Kruzkov inequality (2.1), it is possible to compare u
and u.

3. The nonsti case. 3.1. A{priori estimates. We begin the study of the explicit scheme (1.3) by

collecting some properties of the discrete solutions. Lemma 3.1. Given the conditions of Theorem 2.1, there is a constant M , independent of
t and
x, such that the following a{priori estimates hold for the solution of the explicit scheme (1.3): i) unj 2 S ; ii) kun k1  M iii) TV (un)  M; iv) kun ? umk1  M
tjn ? mj; 80  n; m  N = T=
t: Here, the total variation for grid functions is de ned by

TV (u) :=

X

j 2Z

juj ? uj?1j

and the discrete L1{norm is

kuk1 :=
x

X j 2Z

juj j:

The proof of Lemma 3.1 is given in Section 5. The a{priori estimates of Lemma 3.1 carry over to the entropy solution of (1.1) by a proper application of Helly's theorem cf. Oleinik (1957) or Smoller (1983, chapter 16). This justi es our assumptions on the entropy solution. 3.2. The comparison function. Let us introduce the averaging operator P acting on u 2 L1loc: Z 1 P (u)(x) :=
x u(z) dz for x 2 Ij : I j

3. THE NONSTIFF CASE

6

Now, we de ne the comparison function u
iteratively. It will be convenient to use the arguments y and  instead of x and t and to omit the subscript
. First, we initialize u0(y) = u(y; 0+) = P (u0)(y): Then, we iterate the following three steps for n 2 Z0+. 1) In (tn; tn+1) we solve (3.1) u + f (u)y = 0 with initial data u(y; t+n ). 2) At tn+1 we take cell averages (3.2) u(y; tn+1) = P (u(
; t?n+1))(y); where u(
; t?n ) := lim u(
;  ). !t n  0 there is a constant M independent of step sizes and  such that: i) jL() ? ku(
; T ) ? u(
; t+N )k1 j  M; ii) jR() ? ku0 ? u(
; 0)k1j  M ( +
t); iii) jF ()j  M


t ; 

!

NX ?1 iv) jG()j  M  +
t +
t +
t ku(
; tn) ? u(
; t+n )k1 ;  n=0
t v) jH ()j  M :  Except for iii) and iv), all these estimates carry over from the corresponding estimates for the system cf. Schroll et al. (1994). We rst apply these estimates to complete the proof of Theorem 2.1. From (3.6) we get the error recursion NX ?1
t N 0 e  e + M ( +
t +  ) +
tM en; n=0 p where en := ku(
; tn) ? u(
; t+n )k1. Here, we choose  =
t and use a well known property of the projection operator e0 = ku0 ? u(
; 0)k1 = ku0 ? P (u0)k1  M
t to nd

eN 
tM

NX ?1 n=0

p

en + M
t:

Finally, the discrete Gronwall lemma completes the proof of Theorem 2.1. 3.5. Proof of Lemma 3.2. It remains to prove the bounds iii) and iv) in Lemma 3.2. Here, the a{priori estimates of Lemma 3.1 and the properties of the entropy solution are used. In order to prove iii) note that (u ? k)(f (u) ? f (k)) as a function of k is Lipschitz continous with a Lipschitz constant independent of u: j(u ? k)(f (u) ? f (k)) ? (u ? q)(f (u) ? f (q))j  M jk ? qj; u; k; q 2 S : Using this fact the estimate of F easily carries over from the corresponding estimate in Schroll et al. (1994). Concerning the function G, we have:

jG()j  +

Z TZ X N Z tn Z 0 R n=1 tn?1 ZTX N Z tn Z 0 n=1 tn?1 R

j g(u) ? g(u(y; t+n?1))j ! (x ? y)! (t ?  ) dy d dx dt R

jg(u(y; t+n?1))j j! (t ? tn) ? ! (t ?  )j dy d dt

=: G1() + G2():

4. THE STIFF CASE

10

By using the TV -bound (2.4) and the L1-continuity (2.5) of the entropy solution u we have

jG1()j  Lg

 

0 R n=1 tn?1 R Z TZ Z

ju ? u(y; t+n?1)j ! (x ? y)!(t ?  ) dy d dx dt

ju(x; t) ? u(y; t)j !(x ? y) dy dx dt 0 R R ZTX N Z tn +Lg ku(
; t) ? u(
; tn?1)k1 ! (t ?  ) d dt 0 n=1 tn?1 N Z tn X +Lg k u(
; tn?1) ? u(
; t+n?1)k1 d n=1 tn?1 ZTX N N Z tn X j t ? tn?1 j ! (t ?  ) d dt + M
t en M TV (u(
; t))  + M 0 n=1 tn?1 n=1 Z TZ T N X j t ?  j ! (t ?  ) d dt + M
t en M ( +
t) + M 0 0 n=1 N X M ( +
t) + M
t en: n=1

 Lg



Z TZ X N Z tn Z

Furthermore, G2 is bounded by

jG2()j  Lg

ZTX N Z tn 0 n=1 tn?1

ku(y; t+n?1)k1 j! (t ? tn) ? ! (t ?  )j d dt:

Here, the a{priori estimate ii) of Lemma 3.1 implies

jG2()j  M

ZTX N Z tn Z tn 0 n=1 tn?1 tn?1

j!0 (s ? t)j ds d dt  M
t :

4. The sti case. An interesting point in the arguments developed in the previous

section is that they also apply to the sti equation (1.2). In the case of the semi{implicit scheme (1.4), we have to make sure that there is a unique solution in the state space S := [0; 1]. Furthermore, the generic constant M may not depend on . We have the following a{priori estimates: Lemma 4.1. Given the conditions of Theorem 2.2, the semi{implicit scheme (1.4) has a unique solution satisfying the following estimates: i) ii) iii) iv)

unj 2 S ; TV (un)  TV (u0)  M; kg(un)k1  M kun ? umk1  M
tjn ? mj; 8 0  n; m  N = T=
t:

Here, M is independent of step sizes and .

4. THE STIFF CASE

11

Again, we delay the proof to Section 5 and point out that these a{priori estimates carry over to the entropy solution by Helly's theorem cf. Tveito & Winther (1994). In order to prove Theorem 2.2, we de ne the comparison function as in Section 4, except the pseudo Euler step (3.3) must be replaced by u(y; t+n+1) = u(y; tn+1) +
t g(u(y; t+n+1)): The corresponding Kruzkov{type inequality reads Z TZ

[ju ? kj' + (u ? u)(f (u) ? f (k))'y ] dy d

0 R N Z X

[ju(y; tn) ? kj ? ju(y; t?n ) ? kj]'(y; tn) dy R Zn=1 + [ju0 ? kj'(y; 0) ? ju(y; t+N ) ? kj'(y; T )] dy R N Z X
t [(u(y; t+n ) ? k)g(u(y; t+n ))'(y; tn)] dy ?  n=1 R

+

for all k 2 S and any ' 2 D+ (T ): The same arguments as in the proof of Theorem 2.1 lead to inequality (3.6) with the expressions as before except for G which now reads: Z TZ X N Zt Z G() := 1 (u ? u(y; t+n ))[g(u) !(t ?  ) ? g(u(y; t+n )) !(t ? tn)] 0 R n=1 t ?1 R !(x ? y) dy d dx dt n

n

Z TZ X N Zt Z + + = 1  0 R n=1 t ?1 R(u ? u(y; tn ))[g(u) ? g(u(y; tn ))]!(x ? y)!(t ?  ) dy d dx dt Z Z Z Z N t T X + 1 (u ? u(y; t+n )) g(u(y; t+n )) [! (t ?  ) ? !(t ? tn)] 0 R n=1 t ?1 R !(x ? y) dy d dx dt: The rst term in that sum is negative because of g0 < 0 and therefore ZTX N Zt + ))k j! (t ?  ) ? ! (t ? t )j d dt: k g ( u (
; t G()  1 0  n n 1  n=1 t ?1 Here, the stiness parameter  cancels by the a{priori estimate iii) of Lemma 4.1 G()  M
t : Therefore, we deduce the error estimate  
t N 0 e  e + M  +
t +  p from (3.6). Choosing  =
t concludes the proof of Theorem 2.2. n

n

n

n

n

n

5. APPENDIX

12

5. Appendix. 5.1. Proof of Lemma 3.1. The explicit scheme (1.3) reads unj +1 = uenj +1 +
tg(unj) where uenj +1 is computed by one step of the original Godunov scheme for the homogenous conservation law with data un. Since Godunov's method does not introduce new extrema, we have

juenj +1j  max(junj?1j; junjj; junj+1j) and by (2.3) it follows

junj +1j  max(junj?1j; junjj; junj+1j) +
tLg junjj: Therefore,

kun+1k1  (1 + Lg
t)kun k1  exp(Lg T )ku0k1; con rming i). Furthermore,

kun+1 k1  kuen+1k1 +
tLg kunk1: Since Godunov's method is L1{contracting, we derive that

kun+1k1  (1 +
tLg )kun k1: This implies ii). Concerning the TV {bound iii), we observe that

junj +1 ? unj?+11 j  juenj +1 ? uenj?+11 j +
tLg junj ? unj?1 j: Again, since Godunov's method is total variation diminishing it follows

TV (un+1)  (1 +
tLg )TV (un )  exp(Lg T )TV (u0): Finally, the time{Lipschitz continuity is a consequence of junj +1 ? unjj  juenj +1 ? unjj +
tLg junj j  jf 0(jn )jju(unj; unj+1) ? u(unj?1; unj)j +
tLg junjj where f 0(jn ) = (f (u(unj; unj+1)) ? f (u(unj?1; unj)))=(u(unj; unj+1) ? u(unj?1; unj )). The CFL{condition implies

kun+1 ? unk1 
xTV (un) +
tLg kunk1 = O(
t): Therefore iv) follows by the triangle inequality.

5. APPENDIX

13

5.2. Proof of Lemma 4.1. The solution of scheme (1.4) is implicitly de ned by (5.1)

unj +1 ?
t g(ujn+1) = uenj +1;

where again uenj +1 is computed by one step of the original Godunov method applied to the homogenous equation. The left hand side as a function of u is strictly increasing, therefore, (5.1) has at most one solution. The right hand side is in S = [0; 1] for data in S . Furthermore, since 
t  
t  I ?  g (0) = 0 and I ?  g (1) = 1 ?
t g(1) > 1 there is exactly one solution unj +1 2 S . Concerning the TV {bound, we have for the implicit scheme (1 ?
t g0(jn?+11=2))(unj +1 ? unj?+11 ) = uenj +1 ? uenj?+11 ; where g0(jn?1=2) = (g(unj) ? g(unj?1 ))=(unj ? unj?1). Since g0 < 0, it follows

junj +1 ? unj?+11 j  juenj +1 ? uenj?+11 j and since Godunov's scheme is TVD, the total variation is decreasing. In order to prove iii), we observe g(unj +1) = g(unj) ? g0(jn+1=2)f 0(jn )(u(unj ; unj+1) ? u(unj?1 ; unj)) +
t g0(jn+1=2)g(unj +1); where g0(jn+1=2) = (g(unj +1 ) ? g(unj))=(unj +1 ? unj ). We multiply by (g(unj +1)) and make use of (2.9) to nd (1 +
t )jg(unj +1)j  jg(unj)j + M ju(unj ; unj+1) ? u(unj?1; unj)j: Consequently, by the TV -bound (1 +
t )kg(un+1)k1  kg(un )k1 + M
t: Hence, if M~ = max(M0; M= ) we obtain ~ 0  n  N = T=
t: kg(un)k1  M Finally, iv) is an easy consequence of ii) and iii) because of kun+1 ? unk1  M
tTV (un) +
t kg(un+1)k1 = O(
t):

REFERENCES

14

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