VISCOSITY METHODS FOR PIECEWISE SMOOTH SOLUTIONS TO SCALAR CONSERVATION LAWS. 0. Introduction Consider the single hyperbolic conservation laws

Transcription

1 MATHEMATICS OF COMPUTATION Volue 66, Nuber 28, April 997, Pages S VISCOSITY METHODS FOR PIECEWISE SMOOTH SOLUTIONS TO SCALAR CONSERVATION LAWS TAO TANG AND ZHEN HUAN TENG Abstract. It is proved that for scalar conservation laws, if the flu function is strictly conve, and if the entropy solution is piecewise sooth with finitely any discontinuities which includes initial central rarefaction waves, initial shocks, possible spontaneous foration of shocks in a future tie and interactions of all these patterns, then the error of viscosity solution to the inviscid solution is bounded by Oɛ log ɛ + ɛ inthel -nor, which is an iproveent of the O ɛ upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is iproved to Oɛ.. Introduction Consider the single hyperbolic conservation laws. u t + fu =, <<, t >, subject to the initial condition.2 u, = u, <<. The viscosity ethod approiating the conservation laws. and.2 is to solve the parabolic equation.3 v ɛ t + fv ɛ = ɛv ɛ, subject to the sae initial condition.4 v ɛ, = u. Here, ɛ>isasall paraeter. Viscosity ethods play an iportant role in theoretical analysis on nuerical ethods for hyperbolic conservation laws. A useful technique for studying the behavior of solutions to difference equations is to odel the difference equation by a differential equation called odified equation, see for eaple Chapter of LeVeque s book [3]. The difference equation was introduced because it is easier to solve nuerically than the partial differential equation PDE. This is true if we want to generate nuerical approiations, but on the other hand it is often easier to predict qualitative behavior of a PDE than of a syste of difference equations. Since any odified equations have siilar Received by the editor Noveber 2, 995 and, in revised for, April 5, Matheatics Subject Classification. Priary 65M, 65M5, 35L65. Key words and phrases. Hyperbolic conservation laws, error estiate, viscosity ethods, piecewise sooth. Research of the first author was supported by NSERC Canada Grant OGP5545. Research of the second author was supported by the National Natural Science Foundation of China and the Science Fund of the Education Coission of China. 495 c 997 Aerican Matheatical Society License or copyright restrictions ay apply to redistribution; see

2 496 TAO TANG AND ZHEN HUAN TENG fors to equation.3 ore precise for is equation 6., see Section 6 for ore discussions, a better understanding of the solution behavior of.3 will help us to understand any difference ethods for.. For eistence and regularity properties for.3, we refer to [9, 6]. In this work, we are concerned with the accuracy of the viscosity ethods. More precisely, we will investigate the L -convergence rate of u v ɛ.itiswellknownthat the L -nor of error decays only like O ɛ if the initial data is piecewise sooth. Consider an advection diffusion equation of the for v t + av = ɛv, a = constant, with u = sign. Its eact solution is v ɛ, t = 4 at/ 4ɛt e y2 dy. π Fro this and the fact that the solution to the pure advection proble is u, t = sign at, it can be shown that u,t v ɛ,t L R =C ɛt for soe constant C independent of ɛ and t. This indicates that the L -convergence rate of the viscosity ethods is only one half for this siple linear proble. In practice, solutions of onotone difference ethods agree very well with the true solutions of related odified equations. It is epected that onotone difference ethods will have siilar convergence rate to that of viscosity ethods. Harten, Hyan and La [7] pointed out that onotone difference ethods are of at ost first-order accuracy and Kuznetsov [] provided an one half order L -convergence rate for BV bounded initial data. Tang and Teng [9] recently proved that the best L -convergence rate for onotone difference ethods to scalar conservation laws is one half if it includes the linear flu case. All these indicate that viscosity ethods and onotone difference ethods share soe coon properties on the L -convergence rate. Although it is proved that half order convergence rate is best possible for both viscosity ethods and onotone difference ethods, it is widely believed that these ethods ay have higher order of convergence rate when applying to conve conservation laws. More precisely, if the flu f is strictly conve,.5 f α>, then higher order greater than one half convergence rate is epected, see Harten [6]. Lucier [4] used one of the onotone schees, Godunov s ethod, to solve Rieann probles for. with a conve flu. He observed an order one convergence rate. Teng and Zhang [2] recently proved that for a special class of initial data, naely piecewise decreasing constants, both viscosity and strictly onotone difference ethods have order one L -convergence rate, provided that the flu f is strictly conve. Likewise, Bakhvalov [] and Harabetian [5] proved a faster convergence rate Oɛ ln ɛ for Rieann probles with a rarefaction wave. Teng and Zhang s work used the eistence of traveling waves obtained by Jennings [8]. It is noted that Jennings results do not cover Godunov s ethod since his proof relies heavily on the strict onotonicity of the schees and differentiability of the flu function. In a recent work of Fan [3], the eistence and structure of traveling License or copyright restrictions ay apply to redistribution; see

3 VISCOSITY METHODS FOR CONSERVATION LAWS 497 waves for Godunov s ethod are investigated. The order one L -convergence rate is established for Godunov s ethod on single shock solutions. In this paper, we also consider. with a strictly conve flu, but with a ore general and practical class of initial data. With this class of initial data, the entropy solution to. ay include initial central rarefaction waves, initial shocks, possible spontaneous foration of shocks in a future tie and interactions of all these patterns. We will establish Oɛ ln ɛ L -convergence rate for the piecewise sooth solutions with finitely any discontinuities. To the best of our knowledge, no global error estiates for piecewise sooth solutions have been obtained, although soe results for Rieann probles [, 3, 5], for piecewise constant solutions [2], and for BV bounded initial data [4, ] are available. For piecewise sooth solutions to hyperbolic systes with finitely any noninteracting shocks, Oɛ local error estiates have been obtained by Goodan and Xin [4]. The reainder of the paper consists of si sections. In Section, we investigate the structures of the solution to. with a class of piecewise sooth initial data. Soe useful estiates for first order and second order derivatives of the solution are obtained. In Section 2, we introduce a traveling wave solution and a stability lea, which play iportant roles in obtaining the L -convergence rate. Using the stability lea, we prove in Section 3 the following estiate: sup u,t v ɛ,t L R CT ɛ log ɛ + ɛ, t T provided that u is piecewise sooth and that f u has a finite nuber of inflection points. In our proof, we will use a atching ethod developed by Goodan and Xin [4] and Liu and Xin [2]. In Section 4, we ake soe rearks on optial error estiates. It is shown that if neither initial central rarefaction waves nor spontaneous shocks occur, then sup u,t v ɛ,t L R CTɛ. t T In Section 5, we derive sharp estiates for viscosity ethods with a special class of initial function, which includes the initial data considered in Teng and Zhang [2]. Loosely speaking, we consider a nonincreasing piecewise sooth solution with finitely any shock interactions. In this case, we obtain the following estiate: sup u,t v ɛ,t L R cɛ, t< for soe c independent of ɛ. Soe possible generalizations will be discussed in the final section.. Solution structures Throughout this paper, the nor denotes the standard L -nor, L R; C denotes a positive constant independent of ɛ; c denotes a positive constant independent of t and ɛ, but with different values at different places. Let au =f u. If = is an inflection point of au and also satisfies ȧu <, then we call a decreasing inflection point. If this inflection point satisfies ȧu, we call it an increasing inflection point. Throughout this License or copyright restrictions ay apply to redistribution; see

4 498 TAO TANG AND ZHEN HUAN TENG paper, we use the following notations: ȧu := d au ; äu := d2 d u := du d ; ü := d2 u d 2. d 2 au ; The behavior and structure of entropy solutions for scalar conve conservation laws have been studied for any years, see for eaple [2,, 5, 7]. It is well known that if the initial function is piecewise C 2 -sooth, then the entropy solutions consist of at ost a countable nuber of C 2 -sooth regions. Tador and Tassa [8] proved that if the initial speed has a finite nuber of decreasing inflection points, then it bounds the nuber of future shock discontinuities. We ake the following assuptions on the initial data: A: u is bounded and piecewise C 2 -soothwithafinitenuberof discontinuous points γ i, i I; u γ i ± and u γ i ± eist and are finite; A2: ü L R; A3: li ȧu = ; A4: äu changes signs a finite nuber of ties, i.e. au has a finite nuber of inflection points. Reark.. In order to obtain an estiate for u Lea.3, we have to assue that u is piecewise C 2. The only place to use this assuption is to obtain the estiate.3. Otherwise, a weak assuption on u, i.e. u is piecewise C, will be sufficient. Furtherore, the eistence of ü γ i ± is not required. That is, ü γ i ± can be infinity. We now introduce following notations: Denote the set of points where ȧu has a negative iniu by ζ p, p P. Without loss of generality, we assue that ȧu ζ p are distinct: ȧu ζ < ȧu ζ 2 < <ȧu ζ P. It is straightforward to etend our results to the case when there are soe ζ p,ζ q,p q, such that ȧu ζ p = ȧu ζ q. Denote t =andt p = /ȧu ζ p,p =,,P. For a fied T>t P,we let t P + = T. Reark.2. A negative iniu point ay for a shock at a future tie. However, by no eans will every negative inial point eventually generate a new shock. It is an easier case if the negative inial point does not lead to a shock. Without loss of generality, we assue that each point in the set {ζ p } corresponds to a new shock created at a later tie. The starting point of the new shock is p,t p. Since u is assued to be only piecewise C 2, it ay be discontinuous and therefore will not have a classical derivative. We refer by ȧu to the generalized derivative of au. In the sooth interval γ i,γ i+, ȧu is the classical derivative. In positive and negative jup points of u,ȧu is positive and negative infinity, respectively. In particular, in negative jup points i.e. decreasing discontinuities License or copyright restrictions ay apply to redistribution; see

5 VISCOSITY METHODS FOR CONSERVATION LAWS 499 of u,ȧu has a negative infinite iniu. If ȧu has a continuu of negative inial points it is considered as one iniu. The assuption A4 indicates that ȧu has a finite nuber of decreasing inflection points. It can be shown that under the assuptions A, A3 and A4 only a finite nuber of shocks will occur see, e.g. [7, 8]. The corresponding entropy solution consists of a finite nuber of C 2 -sooth pieces. The shocks are disjoint, ecept for coon endpoints arising fro collisions, and without loss of generality we assue that no ore than two shocks ever collide. More precisely, for each tie interval [t p,t p+ ] there eist a finite nuber of subintervals, t p,n t < t p,n+,n =,,,N p, where t p, = t p,t p,np = t p+, such that in each interval [t p,n,t p,n+ there are a finite nuber of sooth curves = X p,n t,=,,m p,n, satisfying. For t t p,n,t p,n+, X p,n t <Xp,n + t, =,,M p,n ; 2. there eists at least one such that X p,n t p,n+ =X p,n + tp,n+ ; 3. u, t issoothecepton=x p,n 4. ux p,n ±,t eist and satisfy either ux p,n X p,n is a shock, or X p,n,t=ux p,n discontinuity. Moreover, for t t p,n,t p,n+, Ẋ p,n t = fux p,n ux p,n f ux p,n t, t p,n t<t p,n+ ;,t>ux p,n +,t fuxp,n,t +,t ux p,n,t if ux p,n +,t, i.e. X p,n, +,t>ux p,n,t,,t, if ux p,n +,t=ux p,n,t, +,t, i.e. is a contact where Ẋt =dxt/dt; 5. Each shock X p,n t continues to the net subinterval [t p,n+,t p,n+2, with a possibility that it collides with another wave at t = t p,n+.consequently, each shock curve can be etended to t =. Fro the above results it follows that the solution of. and.2 is a finite cobination of the cases plotted in Figure. To proceed, we need ore inforation on shock and rarefaction discontinuities. Shocks will be fored in the following cases: i if there is an initial negative jup; ii if ȧu has a negative iniu in an interval where u is C 2 sooth; iii if u has positive jup at a point, = z say, and ȧu z+ or ȧu z is a negative iniu. A shock ay also occur as a consequence of an interaction of two waves. If = X t and=x + t are interacted with each other, then we denote the resulting shock still as = X t. It is regarded as an etension of = X t or =X + t, rather than a new shock. Consider first two cases. Assue = Xt is a shock curve. Before = Xt is interacted by other waves, say t p <t<t p,n, we can trace two characteristic lines License or copyright restrictions ay apply to redistribution; see

6 5 TAO TANG AND ZHEN HUAN TENG Figure. Solution structures fro = Xt backward to t =:. Xt=ζ +au ζ t = ζ + + au ζ + t, where ζ + t >ζ t. Differentiating both sides of the first identity in. with respect to t gives.2 or equivalently.3 Observe that.4 ζ = Ẋt au ζ +ȧu ζ t, ζ := dζ t, dt +ȧu ζ t = ζ Ẋt au ζ. Ẋt au ζ = = aθu + + θu dθ au f ξθdθu + u, License or copyright restrictions ay apply to redistribution; see

7 VISCOSITY METHODS FOR CONSERVATION LAWS 5 where u ± = uxt ±,t, ξ denotes soe interediate value. Therefore, we have Ẋt au ζ c u + u c au + au = c t ζ+ ζ, where c is soe constant independent of t, and in the last step we have used.. The above results lead to +ȧu ζ t ct ζ ζ + ζ. Graphically, it is easy to see that ζ is a decreasing function of t and ζ + is an increasing function of t. Therefore, we have ζ = ζ ζ + ζ.itisalsotrue that ζ + >ζ. Therefore, we have +ȧu ζ t ct ζ + ζ ζ + ζ. A siilar result holds for ζ +. Consequently, we conclude for both Cases i and ii that +ȧu ζ ± t ct ζ + ζ.5 ζ + ζ, where c is a constant independent of t, and the inequality holds before = Xt collides with another shock. Reark.3. The inequality.5 is an iportant result. Assue that = Xt is not interacted by other waves for t S, T. It follows fro.5 that T.6 S +ȧu ζ ± t dt CT logζ + T ζ T + logζ + S ζ S. For a finite T,wehave ζ + S ζ S ζ + T ζ T C. To bound the right-hand side of.6, we only need to find the lower bound for ζ + S ζ S. See, for eaple, step in the proof of Lea.. In Case iii, a new shock is generated on the boundary of a central rarefaction wave. Since one side of the resulting shock is within the rarefaction wave, one of the two characteristic lines fro any point on the shock will end up at the positive jup point. Consider, for eaple, u has positive jup at = z and ȧu z is a negative iniu in the sense that ȧu z < ȧu for in the left neighborhood of z. There we will have a central rarefaction wave with boundaries = X L t and=x R t. Starting at the point X L t z,t z, where t z = /ȧu z, a new shock = Xt will be generated. It satisfies.7 Now.3 holds, with Ẋt = Xt =ζ +au ζ t. aθu + + θu dθ, License or copyright restrictions ay apply to redistribution; see

8 52 TAO TANG AND ZHEN HUAN TENG where u + = a Xt z/t,u =uxt,t. It follows fro.4 and.7 that Ẋt au ζ c u + u c au + au = c t Xt z au ζ t = c t ζ z. Therefore, we have.8 +ȧu ζ t ct ζ ζ z. Siilarly, if u has positive jup at = z and ȧu z+ is a negative iniu, we have +.9 ct ζ +ȧu ζ + t ζ + z. Reark.4. Siilar to Reark.3, it follows fro.8 that T. S +ȧu ζ t dt CT logζ T z + logζ S z. Therefore, it is iportant to find a bound for ζ S z. See, for eaple, step in the proof of Lea.. A siilar idea will be used to treat.9. Fro the above discussion, we can obtain the following lea which is iportant in the error analysis. Lea.. Assue = Xt is a shock curve satisfying Ẋt = fu+ fu. u + u, u ± := uxt ±,t. Assue au is sufficiently sooth in the neighborhood of its negative iniu points. If = Xt is fored at t =,then T.2 u Xt±,t dt C. If = Xt is fored at t = t p >, then T.3 u Xt±,t dt C log δ + C, t p+δ provided that δ>is sufficiently sall. Here T>t P isafiednuber. Proof. The proof of this lea consists of three steps. Step : Initial negative jup. In this case, a shock is fored due to initial negative jup at a point, say = b. It is obvious that this shock is not going to be interacted by other waves before a finite tie t,n, n N.Itisknownthat u ζ ± +au ζ ± t, t = u ζ ±, <t<t,n. Differentiating both sides with resepect to ζ + or ζ gives.4 u Xt ±,t= u ζ ± +ȧu ζ ± t. License or copyright restrictions ay apply to redistribution; see

9 VISCOSITY METHODS FOR CONSERVATION LAWS 53 Since ȧu ± is bounded by a constant independent of, there eists a constant η such that u Xt±,t cfor all <t<η. If this shock collides with another wave at t = t,n,wechooseapositiveγsatisfying γ<inη, t,n. Obviously,.5 γ u Xt ±,t dt C. Again, since ζ is decreasing and ζ + is increasing, we have ζ + t ζ t ζ + γ ζ γ =constant>, for t γ,t,n. Using Reark.3 and the above inequality, we can obtain that.6 t,n γ u Xt ±,t dt C. At t = t,n, if the shock is interacted by another shock, then the resulting shock satisfies ζ + t ζ t ζ + γ ζ γ =constant>, for t t,n,t,n, where t,n t,n,t is the net possible shock interaction tie. Again, using Reark.3 and the above inequality gives t,n u Xt ±,t dt C..7 t,n At t = t,n, if the shock is interacted by a central rarefaction wave, u, t = a z/t, at t = t,n, say on the left boundary as in Figure G, then we have z ζ t z ζ t,n z ζ γ =constant>, for t t,n,t,n. Using Reark.4 and the fact that u Xt+,t C/t for t t,n,t,n gives the sae estiate as.7. Repeating this procedure a finite nuber of ties, we can obtain.2. Step 2: Negative iniu point in a sooth doain. Assue that ζ p z,z 2 is a negative iniu point of ȧu, and au is sufficiently sooth in z,z 2. Then there is a shock curve, = Xt, fored at p,t p, where p = ζ p + au ζ p t p with t p = /ȧu ζ p. Fro each point away fro the curve, we can trace a characteristic line backward in tie to t =. Using La geoetric condition, we know that these lines will not end up with the discontinuous points. Let = ζ + au ζt denote a characteristic line passing through an, t point. We point out that ζ is a function of and t. Forafiedtiet,itiseasyto see that a larger value of corresponds to a larger value of ζ. Thisgives.8 ζ =+ȧu ζt, ȧu ζ := d au ζ. dζ See also Daferos [2] for a siilar result. Notice that ζ p is a negative iniu point of ȧu ζ. This suggests that there eists a positive integer s such that a k u ζ p =, 2 k 2s, anda 2s+ u ζ p >, where a k u := dk d k au. License or copyright restrictions ay apply to redistribution; see

10 54 TAO TANG AND ZHEN HUAN TENG Moreover, there eists a constant γ> such that.9 <a 2s+ u ζ 2a 2s+ u ζ p, for ζ ζ p <γ. Fro.8 and Taylor s epansion, we obtain +ȧu ζ p t + 2s! a2s+ u ξζ ζ p 2s t, where ξ is between ζ and ζ p. Using the fact that ȧu ζ p = /t p, we obtain that.2 t + t p 2s! a2s+ u ξζ ζ p 2s t. If ζ ζ p <γ, we obtain fro.9 and.2 that ζ ζ p Ct t p 2s, tp <t<t p,n, where t p,n t p+ is the sallest tie level when = Xt collides with other waves t p,n = t p+ if this does not happen. Therefore, we have proved that for all ζ near ζ p, { }.2 ζ ζ p in γ,ct t p 2s, t p <t<t p,n. Tracing two characteristic lines fro = Xt backward to t =, see., we obtain fro.2 { }.22 ζ + t ζ t in γ,ct t p 2s, t p <t<t p,n. Fro.4,.22 and Reark.3, we obtain t p,n u Xt ±,t dt C log δ + C, δ+t p where δ> is sufficiently sall. By using the sae arguent as used in step of this proof, we can etend the above integration to the tie interval t p + δ, T. Step 3: Negative iniu point at a positive jup endpoint. Without loss of generality, we assue that u has a positive jup at = z, andȧu z is a negative iniu. Let X L t be the left boundary of the central rarefaction wave and t p = /ȧu z. Then a shock = Xt isforedatthepoint p,t p, where p = X L t p. Assuing that au ζ is sufficiently sooth as ζ z and using Taylor s epansion, we have { } z ζ t in γ,ct t p 2s, where s is a positive constant. This, together with Reark.4, yield that t,n u Xt,t dt C log δ + C, δ+t p where t,n indicates the sallest tie level when the shock collides with another wave. It is clear that u, t =f z/t for Xt + and hence u Xt+,t C/t. We can then conclude that.3 is true for this case. License or copyright restrictions ay apply to redistribution; see

11 VISCOSITY METHODS FOR CONSERVATION LAWS 55 Reark.5. It is observed fro the proof that if the discontinuities are purely due to the initial negative jup, then the O log δ factor is not included in the estiate for the integral of u, see.2. The optial error estiate sees possible in the case when the solution u contains neither a central rarefaction wave nor an original shock generated at a later tie. This will be investigated in Section 4. Reark.6. Assue = Xt is a shock curve generated at t =ort=t p and = ζ ± + au ζ ± t are the corresponding characteristic lines. We have proved in the above lea that.23 T t p+δ +ȧu ζ ± dt C log δ + C. t The partial derivative u will also be discontinuous at a point z if there is an initial positive jup at z, i.e. u z + > u z. This corresponds to a central rarefaction wave. The case that the rarefaction wave collides with a shock is included in Step of the proof for Lea.. Here we are interested in the central rarefaction wave before it is interacted by a shock. Lea.2. Assue a central rarefaction wave is fored at = z. Let=X L t and = X R t be left and right boundaries of the rarefaction wave, respectively. If ȧu z +is not a negative iniu, then u X R t+,t C, u X R t,t Ct. The above results hold before the rarefaction wave is interacted by a shock. ȧu z +is a negative iniu, then u X R t+,t Ct t p, u X R t,t Ct, for t<t p,wheret p = /ȧu z+. The curve = X R t will becoe a shock after t = t p. Siilar results, based on ȧu z, hold for X L t. Proof. Observe that u, t =f z t, X L t X R t, where X L t =z+f u z t, X R t =z+f u z+t. The above result gives that u X R t,t Ct. Moreover, if ȧu z + orȧu z+ < but not a local iniu, then we can show that u X R t+,t Cbefore = X R t collides with other waves. Now consider the case that ȧu z +< is a local iniu. Then a new shock will be fored at t p = /ȧu z +. Siilar to.4 we have u z+ u X R t+,t= +ȧu z+t, <t<t p. This and the fact that t p = /ȧu z + yield The proof is coplete. u X R t+,t Ct p t, <t<t p. If License or copyright restrictions ay apply to redistribution; see

12 56 TAO TANG AND ZHEN HUAN TENG Reark.7. Let = X R t be the right boundary of a central rarefaction wave. It ay collide with a shock at a certain tie, say t = t. We still denote the resulting shock as X R t. Hence, we have etended the curve = X R t tothewholetie interval [,T]. Using Leas. and.2 gives tp+ δ.24 u X R t ±,t dt C log δ + C. t p+δ Moreover, fro each point on = X R t we can trace a right characteristic line = ζ + + au ζ + t back to t =. Here ζ + = z +beforex R t is interacted by a shock. Fro the proofs for Leas. and.2, we can show that tp+ δ.25 t p+δ +ȧu ζ + dt C log δ + C. t We ay etend = X L t in a siilar way and trace a left characteristic line = ζ + au ζ t back to t =. Here, ζ = z before = X L t collides with a shock. We also have tp+ δ.26 u X L t ±,t dt C log δ + C,.27 t p+δ tp+ δ t p+δ +ȧu ζ dt C log δ + C. t In order to provide a sharp L -error bound, soe properties on second-order derivatives are also needed, e.g. see Sections 3.2 and 3.3. Lea.3. Assue that u satisfies the assuptions A A4 and let au be sufficiently sooth near its negative iniu points. Then the solution to. and.2 satisfies.28 tp+ δ t p+δ provided δ is sufficiently sall. u,t dt C log δ + C, p P, Proof. For any fied tie t t p,t p+, there are finite intervals Z l t,z l+ t the nuber of the intervals can be bounded by a constant independent of t such that the transforation = ζ + au ζt aps Z l t,z l+ t in space to θ l t,θ l+ t in ζ space. The solution u, t is continuously differentiable in Z l,z l+ but not at the end points Z l t andz l+ t. It is observed that in Z l t =in l θ l =, and a l l Z l t =aθ l =. l In the case that Z l t andz l+ t are boundaries of a central rarefaction wave, there eists an integer s such that θ s t =θ s+ t =z s and the solution is of the for u, t =f zs, Z l t Z l+ t, t where Z l+ t Z l t [f u z s + f u z s ]t. It follows fro f α> that [f ] ζ C for ζ C. Hence, we have Zl+ t Zl+ t d u, t d C t 2 = C Z l+ t Z l t t 2 Ct, Z l t Z l t License or copyright restrictions ay apply to redistribution; see

13 VISCOSITY METHODS FOR CONSERVATION LAWS 57 which leads to tp+ Zl+ t t p+δ Z l t u, t ddt C log δ + C. If θ l <θ l+,wehave+ȧu ζt >forζ θ l,θ l+. Differentiating the equation uζ + au ζt, t =u ζ with respect to ζ gives u ζ + au ζt, t + ȧu t = u äu t + ȧu t 2 + ü +ȧu t. It follows fro the above result that Zl+ t Z l t u, t d = θl+ t θ l t θl+ t θ l t u ζ + au ζt, t + ȧu t dζ u äu t θl+ t + ȧu t 2 dζ + θ l t ü +ȧu t dζ =: I + I 2. Observe that äu t = ζ +ȧu t + ȧu t 2. This suggests that the aiu of / + ȧu t possibly occurs at i increasing inflection points of au ; ii decreasing inflection points of au ; and iii the end points θ l t orθ l+ t. For the possibility i, we have / + ȧu t. For the possibility ii, noting that + ȧu t>, we have +ȧu t a. t j>t t/t j Since t t p,t p+, we obtain fro the above result that +ȧu t C t p+ t. Consequently, we obtain for the above three possibilities that.29 +ȧu t + C t p+ t + +ȧu θ l t + +ȧu θ l+ t. Notice that there is no new shock being generated for t t p,t p+. If = Z l t corresponds to a shock curve, then θ l t =ζ + t following our notation in Lea.. Then using Reark.6 we have tp+ dt C log δ + C. t p+δ +ȧu θ l t If = Z l t is a right boundary of a rarefaction wave and collides with a shock at t = t t p,t p+, then using Reark.7 gives tp+ δ dt C log δ + C. t p+δ +ȧu θ l t Siilar results hold for the last ter in.29. These, together with the assuption A2, yield tp+ δ t p+δ I 2 dt C log δ + C. License or copyright restrictions ay apply to redistribution; see

14 58 TAO TANG AND ZHEN HUAN TENG Observe that b äu t c + ȧu t 2 dζ = +ȧu ct +ȧu bt, provided that au C 2 b, c. It follows fro A3, A4 and the above identity that.3 θl+ t äu t θ l t + ȧu t 2 dζ +ȧu θ l tt + +ȧu θ l+ tt + j 2 +ȧu y j t, where y j θ j,θ j+ are inflection points of au. Siilar to the discussion for I 2, we have tp+ δ θl+ t äu t dζdt C log δ + C. t p+δ θ l t + ȧu t 2 Using A and A2, we can show that u is bounded alost everywhere. This and the above inequality yield that tp+ δ t p+δ The proof of Lea.3 is thereby coplete. I dt C log δ + C. 2. Stability and traveling wave Our error estiates are based on a stability lea for nonhoogeneous viscous equations and a traveling wave lea. We first introduce the stability lea. Lea 2.. Let v i, t,i =,2,be continuous and piecewise sooth solutions of the following equations: 2. v i + fv i ɛ v i = g i, t, t > a, i =,2. t The above equation holds for all values of ecept on soe curves X t, M, wherev i ay not eist. If w := v v 2 as,then 2.2 w,t w,a + +ɛ M = Proof. It follows fro 2. that 2.3 t a t a g,τ g 2,τ dτ w X τ+,τ w X τ,τ dτ. w t +f ζw ɛw = g, t g 2, t, where ζ is soe interediate value between v and v 2. If w orw for all, then straightforward integration on the above equation gives 2.2. Let p t <p t<p 2 t< be the points such that at those points w changes signs. License or copyright restrictions ay apply to redistribution; see

15 VISCOSITY METHODS FOR CONSERVATION LAWS 59 Let α j be the sign of w in p j,p j+. Multiplying 2.3 by α j and integrating the resulting equation over p j,p j+ gives pj+ 2.4 α j w t d = ɛ α j w p j+,t α j w p j +,t + p j p j<x <p j+ ɛα j pj+ +α j g, t g 2, t d. p j w X t+,t w X,t Since wp j,t=wp j+,t=andα j w for p j,p j+, we have d pj+ pj+ w d = α j w t d. dt p j p j Moreover, observing that α j w p j+,t andα j w p j +,t, we obtain fro 2.4 that d pj+ w d w X +,t w X,t dt p j ɛ p j<x <p j+ pj+ + p j ɛ p <X <p p g, t g 2, t d. Since the above inequality is true for all j, we have d p w d w X +,t w X,t dt p + p g, t g 2, t d, where p =sup j p j.ifp <, using a siilar ethod as above gives d w d ɛ w X t+,t w X,t dt p X >p + p g, t g 2, t d. In obtaning the last inequality, we have used the fact that w as.using the above two inequalities, we obtain w, t d ɛ t w X +,τ w X,τ dτ p a X >p t + g, τ g 2, τ ddτ + w, a d. a p p A siilar result which replaces p, by,p, respectively, in the above inequality, can be established. These results lead to 2.2. Assue X t is a sooth curve satisfying the Rankine-Hugoniot jup condition: Ẋ t = fu+ fu 2.5 u + u, License or copyright restrictions ay apply to redistribution; see

16 5 TAO TANG AND ZHEN HUAN TENG where Ẋt :=dx/dt is the shock speed, u+ = ux t+,,u = ux t, with u + <u.ifu ± are independent of t, it is shown in [2] that there is a traveling wave solution of the for V ɛ St, where the wave speed S is a constant, satisfying.3 and V ɛ =u,v ɛ =u +. In the case when u± are functions of t, we need the following generalization. Lea 2.2. Let u + t <u tbe two given functions and X t be defined by 2.5. LetVζ;u +,u be defined iplicitly by ζ = V 2 u+ +u [ Φu; u +,u ] du, Φu; u +,u =fu fu Ẋu u. With respect to ζ, the function V ζ; u +,u is a decreasing function satisfying 2.8 V ; u +,u =u, V ; u +,u = 2 u+ +u, V ;u +,u =u +. It also satisfies the following inequalities V ζ; u +,u Hζ;u +,,u u 2.9 u + ep{ αu u + ζ /2}, V ɛ X ;u +,u H X ;u +,u 2. 4 α ɛ, where α is defined by.5. In the above inequalities, V ɛ ζ; u +,u := Vζ/ɛ;u +,u, H is the so-called Heaviside function defined by { H; u +,u = u + if, u if <. Denote V ɛ X t; u +,u byvɛ. Direct calculation shows that 2. V ɛ t + f V ɛ ɛ V ɛ = V ɛ u + u + + V ɛ u u, where V ɛ indicate the partial derivatives for V ɛ with respect to the paraeters u ± ɛ u ±, respectively. Since V H X ; u +,u asɛ +see 2., it is epected that the right-hand side of 2. approaches H X t; u +, u as ɛis sufficiently sall. Lea 2.3. Let V ɛ ζ; u +,u be defined by that in Lea 2.2. We have V ɛ u u ; +,u u V ɛ 2.2 u ;u +,u u H ; u +, u Cɛ u X +,t + u X,t, where u ± = du ± /dt. We defer the proofs of the above two leas to the Appendi. License or copyright restrictions ay apply to redistribution; see

17 VISCOSITY METHODS FOR CONSERVATION LAWS 5 3. Main result and outline of proof In the preceding sections we were concerned with the solution structure, traveling wave solution and stability lea. Using these results, we are now ready to state and prove the ain result of this paper. Theore 3.. Let the flu f be strictly conve. Assue that the initial data u satisfies the requireents A-A4 stated in Section and that au is sufficiently sooth near its negative iniu points. If v ɛ is the solution to.3 and.4 and u is the solution to. and.2, then the following error estiate holds: 3. sup t T v ɛ,t u,t CT where CT is a constant independent of ɛ. ɛ log ɛ + ɛ, Before giving a proof for the above estiate, we point out a direct application of Theore 3.. Corollary 3.. Let the flu f be strictly conve and u C 2 R. If au has a finite nuber of inflection points and is sufficiently sooth at the decreasing inflection points, then the following error estiate holds: 3.2 sup v ɛ,t u,t CT ɛ log ɛ + ɛ. t T Reark 3.. Loosely speaking, the above result iplies that for scalar conve conservation laws, if the initial data u is sooth and f u has a finite nuber of inflection points, then the L -convergence rate of viscosity ethods is Oɛ log ɛ. The class of initial data described in Corollary 3. is quite general, and is widely used in practical coputations. We will use Lea 2. to prove Theore 3.. An essential requireent in Lea 2. is that the solution to 2. should be continuous. It is known that the solution of.3 is continuous for t> if the initial data is piecewise sooth. However, the solution of. ay be discontinuous for t> even if the initial data is sooth. Therefore, we cannot apply Lea 2. directly to equations. and.3. As discussed before, there are two kinds of discontinuity, naely rarefaction wave and shock wave for solutions of. and.2. Across the rarefaction region, the solution is continuous, but on the shock curves, the solution is discontinuous. We need to construct a reasonable approiation to u so that it can get rid of the shock discontinuities. We will consider the cases when u possesses shock discontinuity. For any interval t p,t p+, p P here t =,t P+ = T, we will construct v ɛ,p, t, an approiate function to u, t, such that B: v ɛ,p, t CR t p,t p+, v ɛ,p, t v ɛ, as ± ; B2: v ɛ,p, t is piecewise sooth and satisfies 3.3 v ɛ,p t +f v ɛ,p ɛ v ɛ,p =ḡ, t in its sooth regions; where the right-hand side function satisfies 3.4 tp+ ɛ t p+ɛ ḡ,t dt Cɛ log ɛ + Cɛ; License or copyright restrictions ay apply to redistribution; see

18 52 TAO TANG AND ZHEN HUAN TENG B3: u and v ɛ,p are discontinuous on the sae curves, naely, they are discontinuous on X t, M p,for, t R t p,t p+ see Section for the definition of X ; here for ease of notation we ignore the superscript p and n. Moreover, 3.5 v ɛ,p X t+,t v ɛ,p X t,t = u X t+,t u X t,t ; B4: For t t p,t p+, 3.6 v ɛ,p,t u,t Cɛ. It is known that v ɛ, the solution to.3 and.4, belongs to C R t p,t p+. If v ɛ,p satisfies B, B2 and B3, using Lea 2. gives v ɛ,t v ɛ,p,t v ɛ,t p +ɛ v ɛ,p,t p +ɛ + t +ɛ u X τ+,τ u X τ,τ dτ, t p+ɛ for t t p + ɛ, t p+ ɛ. Using Leas.,.2 and 3.4 gives t t p+ɛ ḡ,τ dτ v ɛ,t v ɛ,p,t v ɛ,t p +ɛ v ɛ,p,t p +ɛ +Cɛ log ɛ + Cɛ, for t t p + ɛ, t p+ ɛ. It follows fro the above inequality and B4 that v ɛ,t u,t v ɛ,t p +ɛ u,t p +ɛ +Cɛ log ɛ + Cɛ, for t t p + ɛ, t p+ ɛ. Since u and v ɛ satisfy the following stability results, we obtain u,s u,s 2 C s s 2, v ɛ,s v ɛ,s 2 C s s 2, v ɛ,t u,t v ɛ,t p u,t p +Cɛ log ɛ + Cɛ, t [t p,t p+ ]. Noting that the above inequality is true for all p P, we obtain that for any t [,T], v ɛ,t u,t v ɛ, u, + Cɛ log ɛ + Cɛ = Cɛ log ɛ + Cɛ. To prove Theore 3., what reains is to construct v ɛ,p, p P, satisfying B B4. We will concentrate on t p,t p+, one of the intervals in P p= t p,t p+. Reark 3.2. Once again we point out that the ain difficulty in using Lea 2. is that the solution u ay be discontinuous. This happens when u has shock discontinuities. If the solution u contains contact discontinuities only i.e. rarefaction waves, the solution is still continuous and Lea 2. can be used directly. Therefore, the construction of v ɛ is to reove shock discontinuities and no efforts need be ade to reove the contact discontinuities. For ease of notation, we denote v ɛ,p by v ɛ in the reainder of this section. License or copyright restrictions ay apply to redistribution; see

19 VISCOSITY METHODS FOR CONSERVATION LAWS Zero shock. In this case, no shock occurs in the doain R t p,t p+. We siply choose v ɛ = u. Since there is no shock discontinuity for u, wehave v ɛ = u CR t p,t p+. It is straightforward to verify that this v ɛ satisfies the requireents B, B3 and B4. It is also obvious that ḡ, t in 3.3 equals ɛu. Using Lea.3 we can verify that this ḡ satisfies 3.4. Therefore, the requireent B2 is also satisfied One shock. In this case, we assue that u is sooth in R t p,t p+, ecept on = X t. Here the function X t satisfies 2.5. As entioned before, Lea 2. cannot be applied directly since u CR t p,t p+. In order to overcoe the difficulty, we introduce the following approiate solution: 3.7 [ v ɛ, t =u, t+ V ɛ X ;u +,u H X ;u + ],u. We will verify that this v ɛ satisfies the requireents B-B4. First, we observe that v ɛ X +,t v ɛ X,t=V ɛ +; u +,u Vɛ ; u +,u =, which indicates that v ɛ CR t p,t p+. It is also easy to see that v ɛ u as ±, which iplies that v ɛ v ɛ as ±. Hence, B is true. We can also show that B3 is true. Using Lea 2.2 will give B4. We now need to show that v ɛ satisfies B2. Proof of B2. We denote V ɛ X ; u +,u byvɛ,andh X ;u +,u by H. It follows fro 2. that V ɛ t + f V ɛ ɛ V ɛ = V ɛ u + u + + V ɛ This, together with., gives v ɛ t +f v ɛ ɛ v ɛ =ḡ, t, t t p,t p+, where ḡ, t =I +I 2 ɛu, I = fu f V ɛ + f v ɛ, I 2 = u + + V ɛ u H X t; u +, u. V ɛ u + u Notice that [H X t; u +,u ] =forallecept on the curve = X t. Away fro = X t, we have I = f v ɛ u + V ɛ f uu f V ɛ V ɛ = f v ɛ f u u + f v ɛ f V ɛ V ɛ = f ξ V ɛ H u + f ξ 2 u H V ɛ =: I + I 2, where ξ and ξ 2 are soe interediate values. For >X t, although there are no shock curves across the interval X,, there ay eist a finite nuber of central rarefaction waves in the interval. Let = Y l t, l L, denote left u u. License or copyright restrictions ay apply to redistribution; see

20 54 TAO TANG AND ZHEN HUAN TENG or right boundaries of the rarefaction waves for t t p,t p+. If >X t but Y l t, u, t =u X t+,t+ u Y l t+,t u Y l t,t + u, td. l X It follows fro the above result that, for >X t, 3.8 u, t u X t+,t + l Siilarly, we can show that, for <X t, 3.9 u, t u X t,t + l u Y l t±,t + u,t. u Y l t±,t + u,t. The above results and 2. lead to I,t Cɛ u X t ±,t + l u Y l t±,t + u,t. An application of Leas.,.2 and.3 yields tp+ ɛ t p+ɛ I,t dt Cɛ log ɛ + Cɛ. Using the facts that [H X t; u +,u ] =,a.e. and V ɛ, we have 3. I 2,t C u H H V ɛ d. For any X t, we can show that u, t H a u z ±,t X t. z R For X t, using 3.8 and 3.9 yields u, t H u X t±,t + u Y l t±,t + u,t X t. l Hence, fro 3. we have I 2,t C u X t±,t + l X t u Y l t±,t + u,t H V ɛ d. Using integration by parts and Lea 2.2, we can show that the integral on the right-hand side can be bounded by Cɛ. Further, using Leas.,.2 and.3 gives tp+ ɛ t p+ɛ I 2,t dt Cɛ log ɛ + Cɛ. License or copyright restrictions ay apply to redistribution; see

21 VISCOSITY METHODS FOR CONSERVATION LAWS 55 We have therefore obtained an estiate for I. An application of Leas. and 2.3 yields tp+ ɛ t p+ɛ I 2,t dt Cɛ log ɛ + Cɛ. Using the estiates for I and I 2, together with Lea.3, we establish the inequality 3.4. Reark 3.3. Fro the definition of ḡ, t, we find that it is necessary to estiate u. Therefore, Lea.3 is useful in obtaining our error estiates Two shocks. In this case, we have two sooth shock curves, X t and X + t, satisfying one of the following possibilities: P: X + t >X tfort [t p,t p+ ], see Case H in Figure ; P2: X + t >X tfort [t p,t p+ andx t p+ =X + t p+, see Case D in Figure ; P3: there eists a tie level t p, t p,t p+ such that there are two shock waves for t t p,t p,, but the two shocks eet at t = t p, and for one shock for t [t p,,t p+, see Cases C and D in Figure. For the first two possibilities, we introduce the following approiate solution: [ v ɛ, t =u, t+ V ɛ X t; u +,u H X t; u + ] 3.,u [ + V ɛ X + t; u + + +,u H X+ t; u ],u. We can verify that v ɛ satisfies the requireents B, B3 and B4. Again, we will show that v ɛ satisfies B2. Proof of B2. We denote V ɛ X t; u +,u andh X t; u +,u by V ɛ and H, respectively. It follows fro 2. that V ɛ t + f V ɛ ɛ V ɛ + V ɛ + t + f V ɛ + ɛ V ɛ + = V ɛ u + + V ɛ u + V ɛ + u V ɛ + u +. u + This, together with., gives where u u + + v ɛ t + f v ɛ ɛ v ɛ =ḡ, t, t t p,t p+, u ḡ, t =I +I 2 ɛu, I = fu f V ɛ f V ɛ + + f v ɛ, I 2 = V ɛ u + u + + V ɛ u u H X t; u +, u + V ɛ + u + u V ɛ + u + u + + H X + t; u + +, u +. License or copyright restrictions ay apply to redistribution; see

22 56 TAO TANG AND ZHEN HUAN TENG We only need to estiate I ; other ters on the right-hand side of 3.2 can be handled by Leas 2.3,.,.2 and.3. Observe that I = f v ɛ u + V ɛ + V ɛ + f uu f V ɛ V ɛ f V ɛ + V ɛ + = f ξ v ɛ uu +f ξ 2 v ɛ V ɛ V ɛ +f ξ 3 v ɛ V ɛ + V ɛ + =: I + I 2 + I 3, where ξ,ξ 2,ξ 3 are soe interediate values. Siilar to the proof of 3.8 and 3.9, we have 3.3 u ±,t u X ±,t + u X + ±,t + u Y l t ±,t + u,t, l for any, t R t p,t p+, where = Y l t, l L, denotes left or right boundaries of possible rarefaction waves. This, together with Lea 2.2 ore precisely 2., leads to tp+ ɛ t p+ɛ Observe that I C u H =: J + J 2. Fro 2.9 we have J 2,t =C Cβ + ep I,t dt Cɛ log ɛ + Cɛ. V ɛ V ɛ + H + X + t β + 2ɛ [ = Cβ + + X +t δt/2 Cβ + [ X + δ/2 Cβ + [V V ɛ δt 2ɛ ; u+,u V ɛ + C V + H + V ɛ d V ɛ ] X +t δt/2 d +ep 3δt 2ɛ ;u+,u d δt β + 4ɛ +β ep V ɛ V ɛ β + δt 4ɛ d ], where β = u u +,β + = u + u+ +,δt=x +t X t. It follows fro 2.9 that δt 3δt V 2ɛ ; u+,u V 2ɛ ;u+,u δt V 2ɛ ;u+,u u + δt β ep β. 4ɛ ] License or copyright restrictions ay apply to redistribution; see

23 VISCOSITY METHODS FOR CONSERVATION LAWS 57 Consequently, we obtain 3.5 } δt δt J 2,t Cβ + β {ep β +ep β +. 4ɛ 4ɛ Consider the following function: t p+ t zt = X + t X t, t [t p,t p+ ]. It is obvious that zt C[t p,t p+. For the first possibility, P, X + t X t > for all t [t p,t p+ ], which indicates that z C[t p,t p+ ]. For the second possibility, P2, we have u + t p+ =u + t p+. It follows fro the definition 2.5 that 3.6 Ẋ t p+ Ẋ+t p+ = f ξθdθ u t p+ u + + t p+, where ξ is an interediate value. Since X + t p+ =X t p+, we have <ζ + + t p+ ζ t p+ =t p+ au t p+ au + + t p+ C u t p+ u + + t p+. This and 3.6 lead to Fro this and L Hospital s rule, we have Ẋ t p+ Ẋ+t p+ C>. li zt = t t p+ Ẋ t p+ Ẋ+t p+ C. This indicates that z C[t p,t p+ ]. Therefore, for both possibilities P and P2, there eists a constant γ>, such that zt γfor all t [t p,t p+ ]. It follows fro the definition of zt that δt=x + t X t Ct p+ t, t p t t p+. Fro this and 3.5, we have J 2,t Cβ + β { ep Cβ + t p+ t/ɛ+ep Cβ t p+ t/ɛ This yields tp+ t p J 2,t dt Cɛ. The estiate for J is a little sipler than that for J 2. It follows fro the facts { az<x+t u u, t H z ±,t X t, if <X + t, C, if X + t, }. License or copyright restrictions ay apply to redistribution; see

24 58 TAO TANG AND ZHEN HUAN TENG that J,t =C u H C a u z ±,t z<x +t +C X + V ɛ X+ d V ɛ C a u z ±,t z<x +t +C V δt/ɛ; u +,u u +. d X t V ɛ X t H d V ɛ Using integration by parts and Lea 2.2, we can show that the last integral can be bounded by Cɛ. Siilar to the proof for J 2,wehave <V δt/ɛ; u +,u u + β ep Cβ t p+ t/ɛ. These results, together with 3.3, yield tp+ ɛ t p+ɛ J,t dt Cɛ log ɛ + Cɛ. Therefore, for both possibilities P and P2, we have tp+ ɛ t p+ɛ I 2,t dt C log ɛ + Cɛ. In a siilar anner we can show that the above estiate holds also for I 3. Hence, we have proved that in cases P and P2 the function v ɛ given by 3. satisfied the requireent B2. If the possibility P3 is the case, naely there are two shock curves, X t and X + t, for t t p,t p,, and one shock, X t, for t t p,,t p+, we construct v ɛ in the following way: 3.7 u, t+ V ɛ H + v ɛ = u, t+ d V ɛ + H +, if t t p,t p,, V ɛ H, if t [t p,,t p+. This case can be regarded as a cobination of possibility P2 analyzed above and the one shock case analyzed in the last subsection. It is then not difficult to verify that this v ɛ satisfies the requireents B-B More shocks. Assue that we have M shock curves X t, M for t t p,t p,, where t p, t p+. These shocks ay eerge and there will be a lesser nuber of shocks for t t p,,t p+. Siilar to the last two subsections, we let 3.8 v ɛ, t =u, t+ M = V ɛ X t; u +,u H X t; u +,u, for, t R t p,t p,. We found that it is straightforward to etend the ethod of proof in the last subsection to show that the function v ɛ given by 3.8 satisfies License or copyright restrictions ay apply to redistribution; see

25 VISCOSITY METHODS FOR CONSERVATION LAWS 59 the requireents B-B4. It is found that the following estiates are particularly useful: M 3.9 v ɛ,t u,t H Cɛ; = V ɛ M 3.2 u ±,t u X t ±,t + u Y l t±,t + u,t ; = l v ɛ,t V ɛ k V ɛ 3.2 k u,t H k V ɛ k + V ɛ H V ɛ k k a u z ±,t Cɛ + Cβ k ep Cβ k t p+ t/ɛ X k <z<x k+ + { } β β k ep Cβ t p+ t/ɛ+ep Cβ k t p+ t/ɛ, k for k M, whereβ =u u+,β k = u k u+ k. HerewehavesetX t=,x M+ t =. For t [t p,,t p+, the nuber of shock curves is less than M and we ay use atheatical induction based on the nuber of shock curves to show that an appropriate v ɛ can be found cf. [2]. Therefore, Theore 3. is proved. Reark 3.4. Under the present fraework, the factor O log ɛ cannotbereoved fro our results. We found the factor is due to the estiates for u and u,see Leas. to.3. In soe special cases, these estiates can be iproved so that optial convergence rates can be obtained. 4. Rearks on sharp estiate In this section, we consider the case when the initial function u satisfies the following conditions: C: u is bounded and piecewise C 2 -sooth with a finite nuber of discontinuous points γ i, i I; u γ i ± eist and are finite; u γ i > u γ i +; C2: ü L R; C3: li ȧu = ; C4: au has no decreasing inflection points, but ay have a finite nuber of increasing inflection points. The assuption C4 iplies that there is no original shock generated after t =. The assuption C iplies that there is no central rarefaction waves fored at t =. Therefore, only discontinuities are initial shocks and the interactions of the initial shocks. Assue = Xt is a shock curve. Following Lea., we can show that T T u Xt ±,t dt C, +ȧu ζ ± dt C. t License or copyright restrictions ay apply to redistribution; see

26 52 TAO TANG AND ZHEN HUAN TENG We now use notations introduced in Section. The interval [,T] can be divided into several subintervals: = t < t < < t N = T. In each interval [t n,t n+, there are a finite nuber of sooth curves = X n t satisfying X n t <Xn + t, =,,M n. Fro each point on = X n t, we can trace two characteristic lines back to t = : X n t=ζ +au ± ζ t. ± Using.29 and the assuption C4 gives +ȧu ζt + +ȧu ζ t + + +ȧu ζ+ t, for ζ + ζ ζ+ with =,,M n, where we have set ζ + =,ζ M =. n+ It follows fro 4.2 that the integral for the right-hand side functions over [,T] can be bounded by a constant C. Following the proof for Lea.3 we can show that 4.3 T u,t dt C. Using and repeating the procedures in the last section yield the following theore. Theore 4.. Let the flu f be strictly conve. Assue that the initial data u satisfies the requireents C-C4 stated above. If v ɛ is the solution to.3 and.4 and u is the solution to. and.2, then the following error estiate holds: 4.4 sup v ɛ,t u,t CTɛ, t T where CT is a constant independent of ɛ. It is easy to show that Oɛ convergence rate is the best possible one for viscosity ethods. The above result indicates that if the solution to. and.2 includes neither central rarefaction waves nor spontaneous shocks, then viscosity ethods yield the optial convergence rate. 5. Results on nonincreasing initial data Throughout this section, c denotes a positive constant independent of t and ɛ, but with different values at different places. We consider a special case when the initial function satisfies the following conditions: D: u is bounded, nonincreasing and piecewise C 2 sooth with a finite nuber of discontinuous points γ < <γ I ; u γ i ± eist and are bounded; D2: ü does not change sign when Λ, with Λ a positive constant; D3: li ȧu = ; D4: äu has no inflection points. Since äu does not change sign in each of its sooth doains, we obtain fro D and D3 that äu L R. The assuption D also indicates that u ± is bounded for Λ. Observe that äu =a u u 2 + a u ü, License or copyright restrictions ay apply to redistribution; see

27 VISCOSITY METHODS FOR CONSERVATION LAWS 52 where a u =f u α>. The above results yield that ü L Λ, Λ. This, together with D2 and D3, iply that ü L R. The assuption D also indicates that u γ i > u γ i +. Since u is nonincreasing, it is known that there eists a tie level T such that there is only one shock after t = T. The solution is sooth away fro this shock for t>t. Fro each point on the shock curve, we can trace two characteristic lines Xt =ζ ± +au ζ ± back to t =. Itisobviousthat 5. ζ t<γ, ζ + t >γ I, T t<. It follows fro D that u ζ u γ, u ζ + u γ I +, T t<, which leads to 5.2 u ζ u ζ + u γ u γ I +=constant>, for T t<.fro.2,.4 and 5.2, we have 5.3 ζ c +ȧu ζ t c>, wherewehaveusedthefactȧu due to D. Since ζ <, we obtain that 5.4 li t ζ t =. Moreover, fro D, D3 and D4, we have 5.5 äu ζ, ζ,γ, and äu ζ, ζ γ I,. We need to prove the following results for soe finite values T,T 2 >T : 5.6 u Xt,t dt c, u Xt+,t dt c; T T 2 ζ t u äu t 5.7 u äu t dζdt c, dζdt c; T + ȧu t 2 T 2 ζ + t + ȧu t 2 ζ t ü 5.8 ü dζdt c, dζdt c. T +ȧu t T 2 ζ + t +ȧu t It follows fro 5.4 that there eists a tie level t = T >T such that ζ t Λ fort T. Fro.4 and 5.3, we obtain 5.9 u Xt,t dt c ζ t u ζ dt T T c u u ζ T c, where in the second inequality we have used the facts that ζ and u ζ. Hence the first inequality in 5.6 is proved. Further, using 5.5 gives ζ t äu t + ȧu t 2 dζ = +ȧu ζ t +ȧu ζ t. It follows fro D2 and D3 that u ζ u ζ for ζ,ζ. License or copyright restrictions ay apply to redistribution; see

28 522 TAO TANG AND ZHEN HUAN TENG The above two inequalities lead to ζ t u äu t + ȧu t 2 dζ u ζ +ȧu ζ t = u Xt,t. This and 5.9 yield the first inequality in 5.7. Using D2 and D3 gives ζ t ü dζ = u ζ, for t T. This and 5.5 yield ζ t ü +ȧu t dζ u ζ +ȧu ζ t = u Xt,t. Consequently, the first inequality in 5.8 is proved. The second inequalities in can be proved in a siilar way. Let T =a{t,t 2 }. Using and following the proofs for Leas. and.3, we can show that 5. T u Xt ±,t dt c, Repeating the proof in Section 3., we can obtain 5. v ɛ,t u,t v ɛ,t u,t +cɛ, T u,t dt c. t [T,, where c is independent of t and ɛ. Since the conditions C C4 include D D4, Theore 4. iplies that 5.2 v ɛ,t u,t CTɛ, Cobining 5. and 5.2 gives the following theore. t [,T]. Theore 5.. Let the flu f be strictly conve. Assue that the initial data u satisfies the requireents D-D4 stated above. If v ɛ is the solution to.3 and.4 and u is the solution to. and.2, then the following error estiate holds: 5.3 sup t< v ɛ,t u,t cɛ, where c is a constant independent of ɛ. 6. Generalizations In this section, we will briefly ention two possible generalizations. The first one is a straightforward etension. The ideas in this paper can be applied to a ore general viscosity equation of the for 6. v ɛ t + fv ɛ = ɛbv ɛ v ɛ, where Bu > for all u R and B C R. We only need to change Φ defined by 2.7 to 6.2 Φu; u +,u =[fu fu Ẋu u ]/Bu. License or copyright restrictions ay apply to redistribution; see