c 2000 Society for Industrial and Applied Mathematics

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1 SIAM J. NUME. ANAL. Vol. 38, No. 5, pp c 2000 Society for Industrial and Applied Matheatics ON THE EGULAITY OF APPOXIMATE SOLUTIONS TO CONSEVATION LAWS WITH PIECEWISE SMOOTH SOLUTIONS TAO TANG AND ZHEN-HUAN TENG Abstract. In this paper we address the questions of the convergence rate for approiate solutions to conservation laws with piecewise sooth solutions in a weighted W 1,1 space. Convergence rate for the derivative of the approiate solutions is established under the assuption that a weak pointwise-error estiate is given. In other words, we are able to convert weak pointwise-error estiates to optialerror bounds in a weighted W 1,1 space. For conve conservation laws, the assuption of a weak pointwise-error estiate is verified by Tador [SIAM J. Nuer. Anal., , pp ]. Therefore, one iediate application of our W 1,1 - convergence theory is that for conve conservation laws we indeed have W 1,1 -error bounds for the approiate solutions to conservation laws. Furtherore, the Oɛ-pointwise-error estiates of Tador and Tang [SIAM J. Nuer. Anal., , pp ] are recovered by the use of the W 1,1 -convergence result. Key words. conservation laws, error estiates, viscosity approiation, optial convergence rate AMS subject classifications. 35L65, 65M10, 65M15 PII. S Introduction. We study the convergence of vanishing viscosity solutions governed by the single conservation law 1.1 u ɛ t + fu ɛ = ɛu ɛ,, t>0,ɛ>0 and subject to the initial condition prescribed at t =0, 1.2 u ɛ, 0 = u 0. We are interested in the convergence rate of u ɛ towards the derivative of the inviscid solution, u, of the corresponding inviscid conservation law 1.3 u t + fu =0, which is subject to the sae initial condition 1.4 u, 0 = u 0., t>0, There has been an enorous aount of papers related to the error estiates for the viscosity or ore general approiations to scalar conservation laws. The ethods of analysis include atching ethod and traveling wave solutions see, e.g., Goodan and Xin [5]; atching the Green function of the linearized proble see, e.g., Liu [13]; weak W 1,1 convergence theory see, e.g., Tador [21]; the Kruzkov-functional ethod see, e.g., Kuznetsov [9]; and energy-like ethods see, e.g., Tador and Tang [22]. The results on error estiates include eceived by the editors Noveber 10, 1999; accepted for publication in revised for June 13, 2000; published electronically Noveber 17, Departent of Matheatics, Hong Kong Baptist University, Kowloon Tong, Hong Kong ttang@ath.hkbu.edu.hk. The research of this author was supported by Hong Kong Baptist University and the esearch Grants Councilof Hong Kong. Departent of Matheatics, Peking University, Beijing , People s epublic of China tengzh@s0.ath.pku.edu.cn. The research of this author was supported by the China State Major Key Project for Basic esearch and Hong Kong Baptist University esearch grant FG/98-99/II-14. Part of this work was carried out while this author was visiting HKBU. 1483

2 1484 TAO TANG AND ZHEN-HUAN TENG for BV entropy solutions to 1.3, an O ɛ convergence rate in L 1 obtained by Kuznetsov [9], Lucier [14], Sanders [18], Cockburn Greaud Yang [3], etc. in the BV-solution space, it is shown by Sabac [17] and Tang and Teng [27] that the L 1 -convergence rate of order O ɛ is optial; for BV entropy solutions, an Oɛ convergence rate in W 1,1 obtained by Tador [21], Nessyahu and Tador [15], Nessyahu Tador Tassa [16], Liu and Warnecke [11], Liu, Wang, and Warnecke [12] etc.; for piecewise sooth solutions for 1.3, an Oɛ convergence rate in L 1 obtained by Bakhvalov [1], Harabetian [6], Teng and Zhang [30], Fan [2], Tang and Teng [28], Teng [29], etc.; for piecewise sooth solutions, an Oɛ convergence rate in the sooth region of the entropy solution obtained by Goodan and Xin [5], Engquist and Sjögreen [4], Tador and Tang [22], [25], etc. The results listed above are concerned with the convergence of the approiate solution itself; essentially nothing is obtained for its derivative. In this work, we will investigate the convergence of the first derivative of the approiate solutions. We will assue that the inviscid solution u has finitely any discontinuities, which is the generic situation, [19], [26]. By properly choosing a weighted function, we will obtain an Oɛ-bound for u ɛ u in a weighted W 1,1 space. More precisely, we will show that the following estiate holds: ρ, t u ɛ u + u ɛ u d Cɛ, where ρ is a distance function to the singular support of u, t. In case that there is only one shock discontinuity = Xt, the above result iplies that u ɛ,t u,t L 1 \[Xt h,xt+h] Chɛ for any given h > 0. If there are finitely any shock curves St = {, t = X k t} K k=1, then we have u ɛ,t u,t L1 \ k [X k t h,x k t+h] Chɛ. In this work, the above estiates are established under the assuption that a weak pointwise-error estiate holds away fro the singular support of the entropy solution u, t. In other words, we are able to convert weak pointwise-error estiates to optial error bounds in a weighted W 1,1 space. It is noticed that such a weak pointwise-error estiate is verified for conve conservation laws by Tador [21]. Furtherore, our W 1,1 -estiate recovers the Oɛ-pointwise error bound obtained by Tador and Tang [22], i.e., u ɛ u, t Chɛ, dist, St h, for any given h>0, where St ={, t = X k t} K k=1. Since our result is obtained in a copletely different way, this work gives an independent check of the result in [22]. There are also any studies on the convergence and stability of the viscosity approiations to the syste of conservation laws, see, e.g., [5], [7], [8], [13], [20], [31]. Kreiss and Kreiss [7] and Kreiss, Kreiss, and Lorenz [8] provide an energyethod approach which is different fro the ones used in this work and [22] for the question of interior regularity. In [13], Liu used a pointwise-error estiate to

3 EGULAITY OF APPOXIMATE SOLUTIONS 1485 study the asyptotic stability of a viscous shock profile. The convergence of viscous solutions to inviscid solutions with one shock is investigated by Goodan and Xin [5] and Yu [31]. In the scalar case, we wish to obtain a clear picture about the pointwise convergence rate, not only for one shock but also for finitely any shock discontinuities. Moreover, we wish to provide soe general ethods which can be applied to other types of approiate schees; see [22], [23]. We close this section by ephasizing two iportant points of this work: 1 Unlike ost of previous work on error estiates, the present paper gives the error bounds for the derivative of the approiate solutions. This result, together with previous results by Tador, etc. for pointwise-error estiates, gives a clear picture about the pointwise convergence rates of approiate solutions to scalar conservation laws. 2 The present results suggest that if a weak pointwise-error estiate can be established for the nonconve conservation laws, then the optial pointwise-error estiate can be obtained for the nonconve case. So far, alost no pointwise-error estiates for nuerical approiations have been obtained for nonconve conservation laws. 2. Preliinaries. For ease of eposition we shall ake the following assuptions in this section: A1 the initial data u 0 is piecewisely C 3 -sooth and is copactly supported; A2 we assue that there eists a sooth curve, = Xt, such that u, t is sooth at any point away fro = Xt; A3 there eists a constant 0 <γ 1 and a constant C T > 0 such that 2.1 u ɛ, t u, t C T ɛ γ for Xt >C T ɛ γ. eark 1. We ake the following rearks and observations: a assuption A1 iplies that u,t L 1 which will be used in the proofs of the net section see the estiate 3.8; also, if we are interested in pointwise-error estiates in a finite doain not too far fro the shock curve, then it is reasonable to assue that u 0 is copactly supported; b although we consider only the case with one shock, the etension to finitely any shocks/rarefaction waves can be carried out by following [22]; see section 4 for ore detail discussion; c assuption A3 is satisfied for conve conservation laws with Lip + - bounded initial data, with γ =1/3 see Tador [21]; it can be iproved to γ =1/2 as discussed in Tador and Tang [22]; however, it is still unclear if A3 holds for nonconve conservation laws; d there is no conveity assuption for the flu function f in this section. At a point on the shock curve = Xt, we have the ankine Hugoniot condition 2.2 X t = fuxt+, 0 fuxt, 0. uxt+, 0 uxt, 0 Also the La geoetrical entropy condition is satisfied [10]: 2.3 f uxt,t X t f uxt+,t. The above properties will be used in the analysis in the following section. Following [22], we introduce a function φ C 2 which satisfies i φ α, 1, ii φ > 0, 0,

4 1486 TAO TANG AND ZHEN-HUAN TENG iii φ 1,, where α 1 is a finite constant. The second requireent above iplies that φ is onotonely decreasing for <0and increasing for >0. More precisely, the distance function φ is required to satisfy φ0=0, 0 <φ αφ for 0, 2.4 φ α, φ k C, k =0, 1, 2, where the constants C are independent of. The functions satisfying the above requireent include φ =1 e 2 α/2. eark 2. Unlike in [22], the distance function φ is now etended to the whole. This enables us to ake a unifor treatent to the weighted error function. In this work, we choose α = γ 1 + 1, where γ is the constant given in assuption A3. In other words, 1/γ+1, 1, φ 1, 1. As to be shown in section 4, we have γ = 1 2 for conve conservation laws. 3. Main results. We define the error function between the viscosity solution and the entropy solution as e, t :=u ɛ, t u, t. The ain results of this paper are given in the following theore. Theore 3.1. Assue that assuptions A1 A3 are satisfied. Then for a weighted distance function φ, φ in 1/γ+1, 1, there eists a positive constant CT independent of ɛ such that 3.1 φ Xt e, t + e, t d CT ɛ for 0 t T. In particular, for any given h>0, there eists a constant CT,h independent of ɛ such that 3.2 u ɛ,t u,t L 1 \[Xt h,xt+h] CT,hɛ for 0 t T. An iediate application of the above theore is to recover the optial pointwiseerror estiate of Tador and Tang [22]. Corollary 3.2. Assue that assuptions A1 A3 are satisfied. Then for a weighted distance function φ, φ in 1/γ+1, 1, 3.3 u ɛ u, t φ Xt = Oɛ. In particular, if, t is away fro the shock discontinuity St ={, t = Xt}, then 3.4 u ɛ u, t Chɛ, dist, St h Soeleas. In order to establish the results in Theore 3.1, we need the following three leas which will lead to a Gronwall inequality for the left-hand side function of 3.1. Moreover, in the reainder of this section, we always denote φ = φ Xt,φ = φ Xt.

5 EGULAITY OF APPOXIMATE SOLUTIONS 1487 Lea 3.3. For a weighted distance function φ, φ in 1/γ+1, 1, and for any F L 1, we have 3.5 f u ɛ Ẋt φ F d C φ F d + Cɛ. Proof. We split the left-hand side of 3.5 into two parts: I 1 + I 2, where I 1 = f u ɛ Ẋt φ Xt F d, Xt ɛ γ I 2 = f u ɛ Ẋt φ Xt F d, Xt ɛ γ where γ is the constant given in A3. It follows fro 2.4 that φ C 1/γ. This result gives that 3.6 I 2 Cɛ. Now for Xt ɛ γ, we use the facts that φ 0, Ẋt >f u +, where u + = uxt + 0,t, to obtain f u ɛ Ẋt f u ɛ f u φ φ + f u f u + φ Cf ɛ γ φ + f u Xtφ using A3 Cf Xtφ + f u Xtφ C Xtφ Cφ. Siilarly, by noting that φ 0 for Xt we can also prove that f u ɛ Ẋt φ Cφ for Xt ɛ γ. The above results lead to I 1 C φf L 1. This, together with 3.6, yields the inequality 3.5. The proof of this lea is coplete. Lea 3.4. For a weighted distance function φ, φ in 1/γ+1, 1, there eists a constant C independent of ɛ such that 3.7 φsgne t e d φ f u ɛ e d + C φ e d + C φ e d + Cɛ. Proof. Direct calculation fro the viscosity equation 1.1 gives t e +fu ɛ fu = ɛe + ɛu. Applying the above result gives φsgne t e d = J 1 + J 2 + J 3,

6 1488 TAO TANG AND ZHEN-HUAN TENG where It is easy to verify that J 1 = J 2 = ɛ J 3 = ɛ φsgne fu ɛ fu d, φsgne e d, φsgne u d. 3.8 J 3 = Oɛ. We now estiate J 2 by integration by parts. It is noted that e,t CXt,. As in La [10], we divide the interval [Xt, into intervals, [Xt, = I t, I t =[p t,p +1 t, with p 0 t =Xt and e changing signs across points p, 1. Assuing that 1 s = sgne, I0t we have φsgne e =p0t= 0, s+ e 0, 1 =p+1t s+ e =pt 0 for 1. It follows fro the above results that Xt 3.10 φsgne e d = p+1t 1 s+ φe d =0 p t = 1 +s p +1t φe p+1t 1 +s φ e d p t p t 0 p+1t 1 +s φ e d, p t where in the last step we have used 3.9. Note that φ = 0 when = p 0 t =Xt and e = 0 when = p t, 1. Using integration by parts for 3.10 leads to φsgne e d p+1t 1 +s φ e d Xt p t = Xt φ e d C u 0 BV = O1, where in the second to last step we have used the facts that u ɛ and u are BV -bounded by u 0 BV. Siilarly, we can show that Xt φsgne e d O1.

7 The above two results yield EGULAITY OF APPOXIMATE SOLUTIONS J 2 Oɛ. Finally, we need to estiate J 1. Observe φ sgne fu ɛ fu d Xt 1 +s φ fu ɛ fu 3.12 = + 1 +s p+1t p t p+1t p t fu ɛ fu φ d. Using the following observation fu ɛ fu = f u ɛ f u u, when = p t, 1, we obtain fro 3.12 that φ sgne fu ɛ fu d Xt = 1 +s φ f u ɛ f p +1 u u p + p+1 ] 1 +s φ [f u ɛ e +f u ɛ f uu d p = p+1 1 φf +s u ɛ f uu d + φ f u ɛ e d p Xt }{{} K p+1 1 +s φ f u ɛ f uu d. p }{{} K 2 Using product rule for the integrand of K 1 gives 3.14 K 1 = K 2 p+1 1 +s φf u ɛ f u u d p p+1 1 +s φf u ɛ f uu d p ] = K 2 φ sgne [f u ɛ e +f u ɛ f uu u d Xt φsgne f u ɛ f uu d Xt K 2 + C φ e d + C φ e d. Xt Xt

8 1490 TAO TANG AND ZHEN-HUAN TENG Cobining the above results, 3.13 and 3.14, gives φ sgne fu ɛ fu d Xt Xt φ f u ɛ e d + C Xt φ e d + C Siilarly, we can show that Xt φ sgne fu ɛ fu d Xt φ f u ɛ e d + C Xt φ e d + C The above two results lead to an estiate for J 1 : 3.15 J 1 φ f u ɛ e d + C φ e d + C Xt Xt φ e d. φ e d. φ e d. The desired inequality 3.7 follows fro the above estiates for J 1,J 2, and J 3. Lea 3.5. For a weighted distance function φ, φ in 1/γ+1, 1, there eists a constant C independent of ɛ such that d 3.16 φ e d C φ e d + C φ e d + Cɛ. dt Proof. It follows fro the viscous equation 1.1 and the conservation law 1.3 that e t = fu ɛ fu + ɛe + ɛu. Using the above equation gives φsgnee t d = φsgne fu ɛ fu d ɛ φsgnee d + ɛ φsgneu d. It can be verified that the last ter above is of order Oɛ, and the second to last ter is bounded by ɛ φsgnee d ɛ φ sgnee d = Oɛ, where in the last step we have used the fact that u and u ɛ are BV -bounded. Using the following observation fu ɛ fu = f u ɛ e + f u ɛ f u u = f u ɛ e + f eu, and the fact that u = O1 for, t away fro the shock curve, we can bound the first ter on the right-hand side of 3.17: φsgne fu ɛ fu d C φ e d + C φ e d.

9 Therefore, we have proved that 3.18 φsgnee t d C EGULAITY OF APPOXIMATE SOLUTIONS 1491 φ e d + C φ e d + Cɛ. Using integration by parts we obtain Ẋtφ e d = Ẋt φsgnee d 3.19 C φ e d. Cobining 3.18 and 3.19 we obtain the desired estiate Proof of Theore 3.1. Having the above three leas, we are ready to prove Theore 3.1. Observe that d φ e d = dt Ẋtφ e d + φ t e d. The above result, together with Lea 3.4, yields d φ e d φ f u ɛ e dt Ẋt d + C φ e d + C φ e d + Cɛ. Since e L 1, we apply Lea 3.3 to obtain d dt φ e d C φ e d + C φ e d + Cɛ. The above result, together with Lea 3.5, leads to d 3.20 φ e + e d C φ e + e d + Cɛ. dt The estiate 3.1 in Theore 3.1 follows iediately fro the above Gronwall inequality. It follows fro 2.4 that for any h>0 there eists a constant ch > 0 such that φ ch as h. The estiate 3.2 follows fro the above observation and 3.1. The proof of Theore 3.1 is coplete Proof of Corollary 3.2. Consider the weighted error function φ Xte, t with, t on the right-hand side of the shock curve, i.e., > Xt. In this case, which leads to φ Xt e, t = φ Xte, t Xt φe d φ ed, φ e d + φ e d. Xt

10 1492 TAO TANG AND ZHEN-HUAN TENG Using integration by parts gives Xt φ e d = Xt Cobining the above two results we obtain φ Xte, t 2 This, together with Theore 3.1, yields φsgne e d Xt Xt φ e d. φ e d 2 φ e d. φ Xte, t Cɛ, >Xt. A siilar result holds for, t on the left-hand side of the shock curve. This copletes the proof of Corollary Application to conve conservation laws. Assue that the flu function f in 1.3 is conve and that there eists only one shock discontinuity for the entropy solution of As in [21], we let Lip + denote the Lip + -seinor [ ] + w wy w Lip + := ess sup, y y where [w] + = Hww, with H the Heaviside function. Owing to the conveity of the flu f, the viscosity solutions of 1.1 satisfy a Lip + -stability condition, siilar to the failiar Oleinik s E-condition, which asserts an a priori upper bound for the Lip + -seinor of the viscosity solution 4.1 u ɛ, t u ɛ y, t y u ɛ,t Lip + 1 u 0 1 Lip + + βt, where u ɛ is the solution of , β is the conveity constant of the flu f, f β; consult, e.g., [21]. The above result suggests that if the initial data do not contain nonlipschitz increasing discontinuities, then the viscosity solution of 1.1 will keep the sae property. The sae is true for entropy solution of It is shown in [21] that with the Lip + initial data, the following pointwise-error bound holds: 4.2 u ɛ, t u, t C 3 ɛ for dist, St 3 ɛ, where St ={, t = Xt}. This result can be further iproved by using the results in [22] and [28]: 4.3 u ɛ, t u, t C ɛ for dist, St ɛ. In other words, assuption A3 holds with γ = 0.5 for conve conservation laws with Lip + initial data Oneshock. It is noted that assuption A2, i.e., the entropy solution has only one shock discontinuity, iplies that u 0 ust be Lip + -stable. Therefore, the theories developed in the last section can be applied to conve conservation laws with one shock discontinuity. We suarize what we have shown by stating the following. Assertion 1. Let u ɛ, t be the viscosity solutions of and u, t be the entropy solution of If the flu function f in 1.3 is conve and the entropy solution has only one shock discontinuity St ={, t = Xt}, then the following error estiates hold:

11 EGULAITY OF APPOXIMATE SOLUTIONS 1493 For a weighted distance function φ, φ in 3, 1, a φ Xt u ɛ u, t + u ɛ u, t d Cɛ, b u ɛ u, t φ Xt = Oɛ. In particular, if, t is away fro the singular support, then for any given h>0 c u ɛ,t u,t L 1 \[Xt h,xt+h] Chɛ, d u ɛ u, t Chɛ, Xt h Finitely any shocks. In this general case, we define the weighted distance function as 4.4 K ρ, t = φ X k t. k=1 We can apply the sae techniques as used in section 3 for the weighted error functions u ɛ, t u, tρ, t and u ɛ, t u, tρ, t. The results in Theore 3.1 and Corollary 3.2 can be etended to these error functions. Following [22], we can etend Assertion 1 and conclude the following. Assertion 2. Let u ɛ, t be the viscosity solutions of and u, t be the entropy solution of If the flu function f in 1.3 is conve and the entropy solution has finitely any shock discontinuities St ={, t = X k t} K k=1, then the following error estiates hold: For a weighted distance function φ, φ in 3, 1, a b K k=1 u ɛ u, t φ X k t u ɛ u, t + u ɛ u, t d Cɛ, K φ X k t = Oɛ. k=1 In particular, if, t is away fro the singular support, then for any h>0 c u ɛ,t u,t L 1 \ k [X k t h,x k t+h] Chɛ, d u ɛ u, t Chɛ, dist, St h. Finally, we point out that unlike previous work for studying the viscous conservation laws the approach used in Tador and Tang [22] does not follow the characteristics but instead akes use of the energy ethod. Therefore, their results for the viscosity approiations can be etended to finite difference ethods [23], [24]. It is seen that the present work also akes use of the sae energy ethod and it is epected that the results in this paper can be etended to other types of approiate solutions such as the onotone difference ethods with sooth nuerical flus. EFEENCES [1] N. S. Bakhvalov, Estiation of the error of nuerical integration of a first-order quasilinear equation, USS Coput. Math. and Math. Phys., , pp

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13 EGULAITY OF APPOXIMATE SOLUTIONS 1495 [29] Z.-H. Teng, First-order L 1 -convergence for relaation approiations to conservation laws, Co. Pure Appl. Math., , pp [30] Z.-H. Teng and P. Zhang, Optial L 1 -rate of convergence for the viscosity ethod and onotone schee to piecewise constant solutions with shocks, SIAM J. Nuer. Anal., , pp [31] S.-H. Yu, Zero dissipation liit to solutions with shocks for systes of hyperbolic conservation laws, Arch. ation. Mech. Anal., , pp