c1992 Society for Industrial and Applied Mathematics THE CONVERGENCE RATE OF APPROXIMATE SOLUTIONS FOR NONLINEAR SCALAR CONSERVATION LAWS

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1 SIAM J. NUMER. ANAL. Vol. 9, No.6, pp , December 99 c99 Society for Industrial and Applied Mathematics 00 THE CONVERGENCE RATE OF APPROXIMATE SOLUTIONS FOR NONLINEAR SCALAR CONSERVATION LAWS HAIM NESSYAHU y AND EITAN TADMOR y Abstract. Let fv " (; t)g ">0 be a family of approimate solutions for the nonlinear scalar conservation law u t + f(u) = 0 with C 0 -initial data. Assume that fv" (; t)g are Lip + -stable in the sense that they satisfy Oleinik's E-entropy condition. It is shown that if these approimate solutions are Lip 0 -consistent, i.e., if kv " (; 0) u(; 0)k Lip 0 () + kv " t + f(v" ) klip 0 (;t) = O("), then they converge to the entropy solution, and the convergence rate estimate kv " (; t) u(; t)k Lip 0 () = O(") holds. Consequently, the familiar L p -type and new pointwise error estimates are derived. These convergence rate results are demonstrated in the contet of entropy satisfying nite-dierence and Glimm's schemes. Key Words. Conservation laws, entropy stability, weak consistency, error estimates,post-processing, nite-dierence approimations, Glimm scheme AMS(MOS) subject classication. 35L65, 65M0,65M5.. Introduction. We are concerned here with the convergence rate of approimate solutions for the nonlinear scalar conservation law, u t + f(u) = 0 with C0-initial data. In this contet we rst recall Strang's theorem which shows that the classical La-Richtmyer linear convergence theory applies for such nonlinear problem, as long as the underlying solution is suciently smooth e.g., [, 5]. since the solutions of the nonlinear conservation law develop spontaneous shock-discontinuities at a nite time, Strang's result does not apply beyond this critical time. Indeed, the Fourier method as well as other L -conservative schemes provide simple countereamples of consistent approimations which fail to converge (to the discontinuous entropy solution), despite their linearized L -stability, e.g., [0, 9]. In this paper we etend the linear convergence theory into the weak regime. The etension is based on the usual two ingredients of stability and consistency. On the one hand, the countereamples mentioned above show that one must strengthen the linearized L -stability requirement. We assume that the approimate solutions are Lip + -stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured here in the Lip 0 -(semi)norm. In we prove for Lip + -stable approimate solutions, that their Lip 0 -convergence rate to the entropy solution is of the same order as their Lip 0 -consistency. The Lip 0 -convergence rate is then converted into stronger L p convergence rate estimates. In particular, we recover the usual L -convergence rate of order one half, and we obtain new pointwise error estimates which depend on the local smoothness of the entropy solution. In 3 we implement these error estimates for nite-dierence approimations, using a nite-element representation which is interesting for its own sake. In 4 we apply these error estimates for the Glimm scheme. Other applications of the current framework to spectral viscosity approimations and various viscosity regularizations can be found in [, 5, ].. Approimate solution. We study approimate solutions of the scalar, genuinely nonlinear conservation law (.) t u(; t) + f(u(; t)) = 0; f00 > 0; with compactly supported initial conditions prescribed at t = 0, (.) u(; t = 0) = u 0 (): y School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv ISRAEL received by editors February 5, 99; accepted for publication (in revised form) January 3, 99. This research was supported in part by NASA Contract No. NAS-8605 while the authors were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia Additional support was provided by Oce of Naval Research Contract No. N J

2 506 HAIM NESSYAHU and EITAN TADMOR Let fv " (; t)g ">0 be a family of approimate solutions of the conservation law (.), (.) in the following sense. DEFINITION. A. We say that fv " (; t)g ">0 are conservative solutions if (.3) v " (; t)d = u 0 ()d; t 0: B. We say that fv " (; t)g ">0 are Lip 0 -consistent with the conservation law (.), (.) if the following estimates are fullled : (i) consistency with the initial conditions (.), (.4a) (.4b) (ii) consistency with the conservation law (.), kv " (; 0) u 0 ()k Lip 0 K 0 " kv " t (; t) + f(v " (; t)) k Lip 0 (;[0;T ]) K T ": We are interested in the convergence rate of the approimate solutions, v " (; t), as their small parameter " # 0. This requires an appropriate stability denition for such approimate solutions. Recall that the entropy solution of the nonlinear conservation law (.), (.) satises the a priori estimate [, ] (.5) ku(; t)k Lip + ku 0 k Lip + + t ; t 0: The case ku 0 k Lip + = is included in (.5), and it corresponds to the eact t decay rate of an initial rarefaction. DEFINITION. We say that fv " (; t)g ">0 are Lip + -stable if there eists a constant 0 (independent of t and ") such that the following estimate, analogous to (.5), is fullled: (.6) sup kv " (; t)k Lip + kv " (; 0)k Lip + + t ; t 0: Remarks.. The case of an initial rarefaction subject to the quadratic u f(u) = u demonstrates that the a priori decay estimate of the eact entropy solution in (.5) is sharp. A comparison of (.6) with (.5) shows that a necessary condition for the convergence of fv " g ">0 is (.7) 0 ; for otherwise, the decay rate of fv " (; t)g (and hence of its "! 0 limit) would be faster than that of the eact entropy solution.. The case > 0 in (.6) corresponds to a strict Lip + -stability in the sense that kv " (; t)k Lip + decays in time, in agreement with the decay of rarefactions indicated in (.5). 3. In general, any a priori bound (.8) kv " (; t)k Lip + Const T < ; 0 t T; is a sucient stability condition for the convergence results discussed below. In particular, we allow for = 0 in (.6), as long as the approimate initial conditions are Lip + -bounded. We remark that the restriction of Lip + -bounded initial data is indeed necessary for convergence, in view of the countereample of Roe's scheme discussed in 3. Unless stated otherwise, we therefore restrict our attention to the class of Lip + -bounded (i.e., rarefaction-free) initial conditions, where (.9) L + 0 := ma(ku 0k Lip +; kv " (; 0)k Lip +) < : Finally, we remark that in case of strict Lip + -stability, i.e., in case (.6) holds with > 0, then one can remove this restriction of Lip + -bounded initial data and our convergence results can be etended to include general L loc-initial conditions. The discussion of this case will be dealt elsewhere. W e letkk Lip ; kk Lip +andkk Lip 0 denote respectively, esssup 6=y ()(y) y ( ^ 0 ; ) k k Lip ; where ^ 0 = R supp :, esssup6=y ()(y) y + and

3 CONVERGENCE RATE OF APPROXIMATE SOLUTIONS 507 We begin with the following theorem which is at the heart of matter. Theorem.. A. Let fv " (; t)g ">0 be a family of conservative, Lip + -stable approimate solutions of the conservation law (.),(.), subject to the Lip + - bounded initial conditions (.9). Then the following error estimate holds (.0a) kv " (; T ) u(; T )k Lip 0 C T kv " (; 0) u 0 ()k Lip 0 + kv " t + f(v " ) k Lip 0 (;[0;T ]) ; where C T ( + L + 0 T ) ; := ma f00 : B. In particular, if the family fv " (; t)g ">0 is also Lip 0 -consistent of order O("), i.e., (.4a),(.4b) hold, then v " (; t) converges to the entropy solution u(; t) and the following convergence rate estimate holds (.0b) kv " (; T ) u(; T )k Lip 0 M T "; M T := (K 0 + K T )( + L + 0 T ) : Proof. We proceed along the lines of []. The dierence, e " (; t) := v " (; t) u(; t), satises the error equation (.) where a " (; t) stands for the mean-value t e" (; t) + [a "(; t)e " (; t)] = F " (; t); a " (; t) = and F " (; t) is the truncation error, a[v " (; t) + ( )u(; t)]d; =0 a() f 0 (); F " (; t) := v " t (; t) + f(v" (; t)) : Given an arbitrary ()W ; 0, we let f " (; t)g 0tT denote the solution of the backward transport equation (.a) " t(; t) + a " (; t) " (; t) = 0; t T; corresponding to the endvalues, (), prescribed at t = T, (.b) " (; T ) = (): (.3) Here, the following a priori estimate holds [, Theorem.] k " (; t)k Lip ep( T t ka " (; )k Lip +d) k()k Lip ; 0 t T: The Lip + -stability of the entropy solution (.5) and its approimate solutions in (.6), provide us with the one-sided Lipschitz upper-bound required on the right-hand side of (.3): (.4) ka " (; )k Lip + ma f00 [kv " (; )k Lip + + ku(; )k Lip +] ma f00 [L + 0 ] + : Equipped with (.3), (.4) we conclude (.5a) k " (; t)k Lip ( + L+ 0 T ) ( + L + 0 t) k()k Lip C T k()k Lip ; 0 t T; C T := ( + L + 0 T ) ;

4 508 HAIM NESSYAHU and EITAN TADMOR and employing (.a) we also have (.5b) k " (; )k Lip[0;T ] jaj ma 0tT k" (; t)k Lip() jaj C T k()k Lip ; jaj := majf 0 j: Of course, (.) is just the adjoint problem of the error equation (.) which gives us (.6) (e " (; T ); ()) = (e " (; 0); " (; 0)) + (F " (; t); " (; t)) L (;[0;T ]): Conservation implies that ^e " 0 R e " (; 0)d = 0 and by (.5a) we nd (.7a) j(e " (; 0); " (; 0))j ke " (; 0)k Lip 0k " (; 0)k Lip ( + L + 0 T ) ke " (; 0)k Lip 0 k()k Lip ; similarly, conservation implies that ^ F " 0 R ;[0;T ] F " (; t)ddt = 0 and by (.5a),(.5b) we nd (.7b) j(f " (; t); " (; t)) L (;[0;T ])j kf " (; t)k Lip 0 (;[0;T ]) k " (; t)k Lip(;[0;T ]) ( + jaj )C T kf " (; t)k Lip 0 (;[0;T ]) k()k Lip : The error estimate (.0a) follows from the last two estimates together with (.6). The Lip 0 -convergence rate estimate (.0b) can be etended to more familiar W s;p loc -convergence rate estimates. The rest of this section is devoted to three corollaries which summarize these etensions. We begin by noting that the conservation and Lip + -stability of v " (; t) imply that v " (; T ) { and consequently that the error, v " (; T ) u(; T ), have bounded variation, (.8) kv " (; T ) u(; T )k BV Const [L + 0 ] + T : We can now interpolate between the BV-regularity (.8) and the Lip 0 -error estimate (.0b), with the help of Sobolev inequality stating that 8s [; p ] we have (e.g., [5, Theorem 9.3]) kd wk W s;p Const kd wk W ; kwk L ; = sp p ; p : Applying the latter to the primitive of v " (; T ) u(; T ) we conclude the following. Corollary... Let fv " (; t)g ">0 be a family of conservative, Lip 0 -consistent and Lip + -stable approimate solutions of the conservation law (.), (.), with Lip + -bounded initial conditions (.9). Then the following convergence rate estimates hold (.9) kv " (; T ) u(; T )k W s;p Const T " sp p ; s p ; p :) The error estimate (.9) with (s; p) = (0; ) yields L convergence rate of order O( p "), which is familiar from the setup of monotone dierence approimations [9, 3]. Of course, uniform convergence (which corresponds to (s; p) = (0; )) fails in this case, due to the possible presence of shock discontinuities in the entropy solution u(; t). Instead, one seeks pointwise convergence away from the singular support of u(; t). To this end, we employ a C0(; )-unit mass mollier of the form () = ( ). The error estimate (.0) asserts that (.0) j(v " (; T ) )() (u(; T ) )()j M T " k d d k L :

5 CONVERGENCE RATE OF APPROXIMATE SOLUTIONS 509 Moreover, if () is chosen so that (.a) k ()d = 0 for k = ; ; : : :; p ; then a straightforward error estimate based on Taylor's epansion yields (.b) j(u(; T ) )() u(; T )j p p! kk L ju(p) j loc ; where ju (p) j loc measures the degree of local smoothness of u(; t), ju (p) j loc := k p p u(; T )k L loc (+supp) : The last two inequalities imply the following corollary. Corollary.3.. Let fv " (; t)g ">0 be a family of conservative, Lip 0 -consistent and Lip + -stable approimate solutions of the conservation law (.), (.), with Lip + -bounded initial conditions (.9). Then, for any p-order mollier () ( ) satisfying (.a), the following convergence rate estimate holds (.) j(v " (; T ) )() u(; T )j Const T ( + ju(p) j loc ) " p p+ ; " p+ : p! Corollary.3 shows that by post-processing the approimate solutions v " (; t), we are able to recover the pointwise values of u(; t) with an error as close to " as the local smoothness of u(; t) permits. A similar treatment enables the recovery of the derivatives of u(; t) as well, consult [, 4]. The particular case p = in (.), deserves special attention. In this case, post-processing of the approimate solution with arbitrary C0-unit mass mollier (), gives us (.3) j(v " (; T ) )() u(; T )j Const ( + ju (; T )j loc ) 3p "; 3p ": We claim that the pointwise convergence rate of order O( 3p ") indicated in (.3) holds even without post-processing of the approimate solution. Indeed, let us consider the dierence v " (; T ) (v " (; T ) )() = = y y [v " (; T ) v " ( y; T )] (y)dy = v " (; T ) v " ( y; T ) y y ( y )dy: By choosing a positive C 0-unit mass mollier () supported on (; 0) then, thanks to the Lip + -stability condition (.6), the integrand on the right does not eceed Const, and hence (.4a) v " (; T ) (v " (; T ) )() Const : Similarly, a dierent choice of a positive C 0-unit mass mollier () supported on (0; ) leads to (.4b) v " (; T ) (v " (; T ) )() Const : Each of the last two inequalities (with 3p ") together with (.3) show that the approimate solution itself converges with an O( 3p ")-rate, as asserted. We summarize what we have shown by stating the following. Corollary.4.. Let fv " (; t)g ">0 be a family of conservative, Lip 0 -consistent and Lip + -stable approimate solutions of the conservation law (.), (.), with Lip + -bounded initial conditions (.9). Then the following convergence rate estimate holds: (.5) jv " (; T ) u(; T )j Const ;T 3p "; Const ;T + ju (; T )j L ( 3 p ";+ 3p "):

6 50 HAIM NESSYAHU and EITAN TADMOR Remark. The above derivation of pointwise error estimates applies in more general situations. Consider, for eample, a family of approimate solutions, fv " (; t)g ">0 which satises a standard L (rather than Lip 0 ) error estimate (.6) j(v " (; T ) u(; T ); ())j Const T p "kk L : Then our previous arguments show how to post-process v " (; T ) in order to recover the pointwise values of the entropy solution, u(; T ) with an error as close to p " as the local smoothness of u(; T ) permits. In particular, using (.6) with a positive C0 -unit mass mollier, () = ( ) we obtain p j(v " " (.7) (; T ) )() (u(; T ) )()j Const T kk L : Using this together with we nd (.8) j(u(; T ) )() u(; T )j kk L ku (; T )k L loc (+supp) ; j(v " (; T ) )() u(; T )j Const T ( + ju (; T )j loc ) 4p "; 4p ": If the approimate solutions fv " (; t)g ">0 are also Lip + -stable, then we may augment (.8) with (.4) to conclude the pointwise error estimate (.9) jv " (; T ) u(; T )j Const ;T 4p "; Const ;T + ju (; T )j L ( 4 p ";+ 4p ") : 3. Finite-Dierence approimations. We want to solve the conservation law (.) - (.) by difference approimations. To this end we use a grid ( := ; t n := nt) with a ed mesh-ratio t = Const. The approimate solution at these grid points, vn v( ; t n ), is determined by a conservative dierence approimation which takes the following viscosity form, e.g., [7] (3.) v n+ = v n [f(vn +) f(v n )] + [Qn + (v n v+ v n ) Q n (v n v n )]; n 0; and is subject to Lip + -bounded initial conditions, (3.) v 0 = + u 0 ()d; L + 0 = ku 0k Lip + < : Let v " (; t) be the piecewise linear interpolant of our grid solution, v " ( ; t n ) = v n, depending on the small discretization parameter " # 0. It is given by (3.3) v (; t) = X j;m v m j m j (; t); m j (; t) := j () m (t); where j () and m (t) denote the usual `hat' functions, j () = min( j; j+ ) + ; m (t) = t min(t tm ; t m+ t) + : To study the convergence rate of v (; t) as # 0, we rst have to verify the conservation and the Lip 0 -consistency of the dierence approimation. To this end we proceed as follows. We rst note that v (; t) are clearly conservative, for by the choice of the initial conditions in (3.), v (; t)d = X v n + v n v+ = X v 0 + v 0 + = u 0 ()d:

7 CONVERGENCE RATE OF APPROXIMATE SOLUTIONS 5 Moreover, these initial conditions are Lip 0 -consistent { in fact the following estimate which is left to the reader holds, (v (; 0) u 0 (); ()) Const () ku 0 ()k BV k()k Lip : Finally, we turn to consider the Lip 0 -consistency with the conservation law (.). To this end we compare v (; t) with certain entropy conservative schemes constructed in [8]. A straightforward computation (carried out in the Appendi) shows that there eists a bounded piecewiseconstant function, D n () = P j Dn j+ j+ (), ( j+ () := characteristic function of ( j; j+ )), such that the dierence approimation (3.) recast into the equivalent form (3.4) = t Hence, for arbitrary C 0 (3.5) t v (; t); n (; t) t v (; t); t n (; t) ;t + ;t, we may rewrite (3.4) as f(v (; t)); n (; t) (v t + f(v ) ; ) ;t = ;t v (; t); n (; t) 4X k= T k : = : D();t The sum on the right-hand side of (3.5) represents the truncation error of the dierence approimation (3.), and according to (3.4), it consists of the following four contributions (here, ^(; t) = P ;n ( ; t n ) n (; t) denotes the piecewise-linear interpolant of (; t)): T = (v ; ^ ) D();t ; T = t (v t ; ^ t ) ;t ; T 3 = (vt ; ) ;t (vt ; ^) ;t ; T 4 = (f(v ) ; ) ;t (f(v ) ; ^) ;t : We want to show that the dierence approimation (3.) is consistent with the conservation law (.), in the sense that the Lip 0 -size of its truncation error is of order O(). The required estimates in this direction are collected below. We begin with a straightforward estimate of the rst term, (3.6a) jt j kv k L (jd()j;t) k^ k L (;t) C kv (; t)k L ([0;T ];BV ()) k(; t)k Lip(;[0;T ]) : The dierence approimation (3.) enables us to upper bound time-dierences in terms of spatial dierences to yield the following upper-bound on the second term, (3.6b) jt j t kv t k L (;t) k^t k L (;t) C kv (; t)k L ([0;T ];BV ()) k(; t)k Lip(;[0;T ]) : The third contribution to the truncation error we rewrite as R T 3 = [(v t ; ^) ;t (vt ; ^) ;t ] + (vt ; ^) ;t = T 3 + T 3 : R The Euclidean and weighted L -inner products are denoted by (; ) = () ()d and (; ) D() = () ()D()d. The corresponding discrete P P `- inner product reads (; ) = ( ) (). Similar notations R R P are used for (; t)-functions, e.g., (; ) D();t = n (; tn ) (; t n )D()dt, kk p L p (;t) = t j( ; t)jp dt, etc.

8 5 HAIM NESSYAHU and EITAN TADMOR We have (abbreviating m j ( j ; t m )): T 3 = X ;j;n X v n ) n j ( (); j ()) (vn+ v n ) n ;j;n (vn+ and hence T 3 is upper bounded by X = (vn+ v n ) 6 (n + n + n ); ;n jt 3 j Const:kv t k L (;t) k(; t)k Lip(;[0;T ]) : This together with the standard interpolation error estimate jt 3 j kvt k L (;t) k ^k L (;t) Const kvt k L (;t) k(; t)k Lip(;[0;T ]) ; give us that the third term does not eceed (3.6c) jt 3 j Const kv t k L (;t)k(; t)k Lip(;[0;T ]) C 3 kv (; t)k L ([0;T ];BV ()) k(; t)k Lip(;[0;T ]) : A similar treatment of the fourth term implies (3.6d) jt 4 j C 4 kv (; t)k L ([0;T ];BV ()) k(; t)k Lip(;[0;T ]) : Equipped with the last four estimates (3.6a) - (3.6d), we return to (3.5), obtaining (3.7) j(v t + f(v ) ; ) j Const kv (; t)k L ([0;T ];BV ()) k(; t)k Lip(;[0;T ]) : This shows that the Lip 0 -consistency estimate (.4b) holds with " = and K T kv (; t)k L ([0;T ];BV ()). Thus, Corollaries.-.4 apply and their various error estimates are put together in the following. Theorem 3... Assume that the dierence approimation (3.)-(3.) is Lip + -stable in the sense that the following one-sided Lipschitz condition is fullled: (3.8) L + n [L + 0 ] + t n ; 0 tn T; L + n kv (; t n )k Lip + = ma (v n + v n ) + : Then the following error estimates hold: (3.9a) kv (; T ) u(; T )k W s;p Const T () sp p ; s p ; p ; (3.9b) jv (; T ) u(; T )j Const ;T [ + ma jj 3p ju (; T )j] 3p : Eamples. The following rst order accurate schemes (identied in a decreasing order according to their numerical viscosity coecient, Q + Q n + ), are frequently referred to in the literature. (3.0a) La F riedrichs scheme : Q LF + ; (3.0b) Engquist Osher scheme : Q EO = + v n + vn v+ v jf 0 (v)jdv;

9 (3.0c) CONVERGENCE RATE OF APPROXIMATE SOLUTIONS 53 f(v Godunov scheme : Q G n = ma + ) + f(v n ) f(v) + v v n + ; vn (3.0d) Roe scheme : Q R + = ja + j; a + an + = f(vn +) f(v n ) v n + : vn We comment briey on the Lip + -stability condition of these schemes. The solution of the La-Friedrichs scheme satises [6, Eqn. 3.8] L + n+ L+ n ( tl+ n ); which in turn implies (by induction) the desired Lip + -stability (3.8) with =, for (3.) L + n L+ n ( L+ n ) < [L + n ] + t : : : [L + 0 ] + t n : A similar framework was used in [7, 3] to show that Godunov scheme satises the Lip + -stability estimate (3.8) with = 4. The Lip + -stability of the Engquist-Osher (E{O) scheme is closely related to Godunov's scheme. The two schemes coincide ecept for sonic shock cells (where a(v + ) < 0 < a(v v )), which leads to (3.) ja + j Q G + Q EO + Q G + Hence, the forward dierences of E-O scheme are upper-bounded by Const (v + ) : v n+ + = ( Q EO + )v n + + (QG + 3 a + 3 )vn (QG + a )v n (3.3) + (QEO + 3 Q G + 3 )v n (QEO Q G )v n ( Q EO + )v n + + (QG + 3 a + 3 )vn (QG + a )v n : We distinguish between two cases. If v n 0, then the rst term on the right of (3.3) does not eceed + (Q G + )v n, and hence the E-O solution satises the one-sided Lipschitz condition in this case, because + Godunov's solution does. Otherwise, v n and therefore ( + Q EO )v n is negative, hence + + v n+ + (QG + 3 a + 3 )v n (QG + a )v n and the Lip + -bound follows in view of (3.) and the CFL condition majf 0 j <. Finally, for the Roe (or Courant-Isaacson-Rees) scheme, Lip + -stability (3.8) with = 0 (no decay), was proved in []. Note that the assumption of Lip + -bounded initial conditions is essential for convergence to the entropy solution in this case, in view of the discrete steady-state solution, v 0 = sgn( + ), which shows that convergence of Roe scheme to the correct entropy rarefaction fails due to the fact that the initial data are not Lip + -bounded. Using Theorem 3. we conclude the following. Corollary 3... Consider the conservation law (.), (.) with Lip + -bounded initial data (.9). Then the Roe, Godunov, Engquist-Osher, and La-Friedrichs dierence approimations (3.), (3.0) with discrete initial data (3.) converge, and their piecewise-linear interpolants v (; t), satisfy the convergence rate estimates (3.9a), (3.9b).

10 54 HAIM NESSYAHU and EITAN TADMOR 4. Glimm scheme. We recall the construction of Glimm approimate solution for the conservation law (.), see [6, 4]. We let v(; t) be X the entropy solution of (.) in the slab t n t < t n+ ; n 0, subject to piecewise constant data v(; t n ) = v n (). To proceed in time, the solution is etended (in a staggered fashion) with a jump discontinuity across the lines t n+ ; n 0, where v(; t n+ ) takes the piecewise constant values (4.) v(; t n+ ) = X v n+ + + (); v n+ = v( r n ; t n+ 0): Notice that in each slab, v(; t) consists of successive noninteracting Riemann solutions provided the CFL condition, maja(u)j is met. This denes the Glimm approimate solution, v(; t) v" (; t), depending on the mesh parameters " = t, and the set of random variables fr n g, uniformly distributed in [ ; ]. In the deterministic version of the Glimm scheme, Liu [0] employs equidistributed rather than random sequence of numbers fr n g. We note that in both versions, we make use of eactly one random or equidistributed choice per time step (independently of the spatial cells), as was rst advocated by Chorin [3]. It follows that both versions of Glimm scheme share the Lip + -stability estimate (.6). Indeed, since the solution of a scalar Riemann problem remains in the conve hull of its initial data, we may epress v n+ as + ( n + )v n + n + v n + for some n + [0; ], and hence v n+ + v n+ = n + v n + ( + n )v n : We now distinguish between two cases. If either v n or v n + is negative, then (4.) v n+ v n+ ma(v n + + ; v n ): Otherwise when both v n and + v n are positive, the two values of v n+ and v n+ are obtained as + sampled values of two consecutive rarefaction waves, and a straightforward computation shows that their dierence satises (4.). Thus in either case, the Lip + -stability (.6) holds with = 0. Although Glimm approimate solutions are conservative \on the average," they do not satisfy the conservation requirement (.3). We therefore need to slightly modify our previous convergence arguments in this case. We rst recall the truncation error estimate for the deterministic version of Glimm scheme [8, Theorem 3.], (4.3) v t + f(v ) ; (; t) L (;[0;T ]) Const T h pj ln j kkl + k(; t)k Lip(;[0;T ]) i : Let (; t) = (; t) denote the solution of the adjoint error equation (.). Applying (4.3) instead of (.7b) and arguing along the lines of Theorem (.), we conclude that Glimm scheme is Lip 0 -consistent (and hence has a Lip 0 -convergence rate) of order p j ln j, (4.4) j(e (; T ); ())j Const T h pj ln j kkl + k()k Lip i : To obtain an L -convergence rate estimate we employ (4.4) with yielding pj j(e (4.5) (; T ); )j Const T ln j + k()k L : Using this estimate together with (e " (; T ); [() ()]) (e " (; T ) e " (; T ); ) Const ke " (; T )k BV kk L ;

11 CONVERGENCE RATE OF APPROXIMATE SOLUTIONS 55 imply (for p ), the usual L -convergence rate of order O( p j lnj). As noted in the closing remark of, the Lip + -stability of Glimm's approimate solutions enables us to convert the L -type into pointwise convergence rate estimate. We close this section by stating the following. Theorem 4... Consider the conservation law (.), (.) with suciently small Lip + -bounded initial data (.9). Then the (deterministic version of) Glimm approimate solution v (; t) in (4.) converges to the entropy solution u(; t), and the following convergence rate estimates hold: (4.6) kv (; T ) u(; T )k L Const T p j lnj; (4.7) jv (; T ) u(; T )j Const ;T [ + ma jj 4p ju (; T )j] 4p j lnj: Remarks.. A sharp L -error estimate of order O( p ) can be found in [], improving the previous error estimates of [8].. Theorem 4. hinges on the truncation error estimate (4.3) which assumes initial data which suciently small variation [8]. Etensions to strong initial discontinuities for Glimm scheme and the front tracking method can be found in [4, Theorems 4.6 and 5.]. APPENDIX. We want to show that the piecewise-linear interpolant v (; t) in (3.3) serves as an approimate weak solution of the conservation law (.). Let v n () = X v n () and v (t) = X n v n n (t) denote the spatial and temporal interpolants of the discrete grid solution fv n g ;n. Straightforward integration by parts yields [8] (a) (f(v (; t)) ; ()) = [f(v +(t)) f(v (t))] [Q + (t)v + (t) Q (t)v (t)] where ( we abbreviate v + (; t) [v (t) + v + (t)] + v + (t)) (b) Q (t) = + v + (t) = In particular, for f(v) v we have Q 0 and (a) yields ( 4 )f 00 (v + (; t))d: (v (; t); ()) = [v +(t) v (t)]: Echanging the role of the and t variables in the last equality we get () (vt (; t); n (t)) t = [vn+ () v n ()]: P Moreover, with D() = D + + () we have (3) (v (; t); ( ()) ) D() = [D + v + (t) D v (t)] and by echanging the role of the and t variables in (3) we get (4) t (v t (; t); ( n (t)) t ) t = [vn+ () v n () + v n ()]:

12 56 HAIM NESSYAHU and EITAN TADMOR The equalities () - (4) imply ( 0 ) (f(v (; t)) ; n (; t)) ;t = = t [f(vn +) f(v)] n t [Q + (t n ) n + Q (t n )v n ]; ( 0 ) (vt (; t); n (; t)) ;t = [vn+ v n ]; (3 0 ) (v (; t); ( n (; t)) ) D();t = t [D + vn + D vn n ]; (4 0 ) t (v t (; t); ( n (; t)) t) ;t = [vn+ v n vn ]: The dierence approimation (3.) reads (5) [v n+ v n ] = t [f(vn +) f(v)] n + [Qn + v n + Q n v n ]: By ( 0 ) and (4 0 ), the left-hand side (LHS) of (5) equals LHS = [vn+ v n ] + [vn+ v n + vn ] = Net, we set D n + Qn + = (vt (; t); n (; t)) ;t t (v t (; t); ( n (; t)) t ) ;t : Q + (t n ); then by ( 0 ) and (3 0 ) the right-hand side (RHS) of (5) equals and (3.4) now follows. RHS = t [f(vn + t f(vn )] + [Q + v n + Q v n ]+ + + t [Dn + v n + D n v n ] = = (f(v (; t)) ; n (; t)) ;t (v (; t); ( n (; t)) ) D();t REFERENCES [] Y. Brenier, Roe's scheme and entropy solution for conve scalar conservation laws, INRIA Report 43, France 985. [] Y. Brenier and S. Osher, The discrete one-sided Lipschitz condition for conve scalar conservation laws, SIAM J. Numer. Anal., 5 (988), pp [3] A. J. Chorin, Random choice solution of hyperbolic systems, J. Comp. Phys., (976, pp [4] I. L. Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for ows with strong discontinuities, Comm. Pure Appl. Math., XLII (989), pp [5] A. Friedman, Partial Dierntial Equations, R. E. Krieger Publishing Company, Huntington, New-York, 976. [6] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 8 (965), pp [7] J.B. Goodman and R.J. LeVeque, A geometric approach to high resolution TVD schemes, SIAM J. Numer. Anal., 5 (988), pp [8] D. Hoff and J. Smoller, Error bounds for the Glimm scheme for a scalar conservation law, Trans. Amer. Math. Soc., 89 (988), pp [9] N.N. Kuznetsov, On stable methods for solving nonlinear rst order partial dierntial equations in the class of discontinuous solutions, Topics in Num. Anal. III, Proc. Royal Irish Acad. Conf. Trinity College, Dublin (976), pp

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