Viscosity methods for piecewise smooth solutions to scalar conservation laws

Authors:
Tao Tang and Zhen-huan Teng

Journal:
Math. Comp. **66** (1997), 495-526

MSC (1991):
Primary 65M10, 65M05, 35L65

MathSciNet review:
1397446

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Abstract: It is proved that for scalar conservation laws, if the flux function is strictly convex, and if the entropy solution is piecewise smooth with finitely many discontinuities (which includes initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time and interactions of all these patterns), then the error of viscosity solution to the inviscid solution is bounded by in the -norm, which is an improvement of the upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to .

**1.**N. S. Bahvalov,*Error estimates for numerical integration of quasilinear first-order equations*, Z. Vyčisl. Mat. i Mat. Fiz.**1**(1961), 771–783 (Russian). MR**0152143****2.**C. M. Dafermos,*Generalized characteristics and the structure of solutions of hyperbolic conservation laws*, Indiana Univ. Math. J.**26**(1977), no. 6, 1097–1119. MR**0457947****3.**H. Fan,*Existence of discrete traveling waves and error estimates for Godunov schemes of conservation laws*, Preprint (1996).**4.**Jonathan Goodman and Zhou Ping Xin,*Viscous limits for piecewise smooth solutions to systems of conservation laws*, Arch. Rational Mech. Anal.**121**(1992), no. 3, 235–265. MR**1188982**, 10.1007/BF00410614**5.**Eduard Harabetian,*Rarefactions and large time behavior for parabolic equations and monotone schemes*, Comm. Math. Phys.**114**(1988), no. 4, 527–536. MR**929127****6.**Amiram Harten,*The artificial compression method for computation of shocks and contact discontinuities. I. Single conservation laws*, Comm. Pure Appl. Math.**30**(1977), no. 5, 611–638. MR**0438730****7.**A. Harten, J. M. Hyman, and P. D. Lax,*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**0413526****8.**Gray Jennings,*Discrete shocks*, Comm. Pure Appl. Math.**27**(1974), 25–37. MR**0338594****9.**Heinz-Otto Kreiss and Jens Lorenz,*Initial-boundary value problems and the Navier-Stokes equations*, Pure and Applied Mathematics, vol. 136, Academic Press, Inc., Boston, MA, 1989. MR**998379****10.**N. N. Kuznecov,*The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation*, Ž. Vyčisl. Mat. i Mat. Fiz.**16**(1976), no. 6, 1489–1502, 1627 (Russian). MR**0483509****11.**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653****12.**Jian-Guo Liu and Zhou Ping Xin,*𝐿¹-stability of stationary discrete shocks*, Math. Comp.**60**(1993), no. 201, 233–244. MR**1159170**, 10.1090/S0025-5718-1993-1159170-7**13.**Randall J. LeVeque,*Numerical methods for conservation laws*, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. MR**1153252****14.**Bradley J. Lucier,*Error bounds for the methods of Glimm, Godunov and LeVeque*, SIAM J. Numer. Anal.**22**(1985), no. 6, 1074–1081. MR**811184**, 10.1137/0722064**15.**O. A. Oleĭnik,*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737****16.**Joel Smoller,*Shock waves and reaction-diffusion equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR**688146****17.**David G. Schaeffer,*A regularity theorem for conservation laws*, Advances in Math.**11**(1973), 368–386. MR**0326178****18.**Eitan Tadmor and Tamir Tassa,*On the piecewise smoothness of entropy solutions to scalar conservation laws*, Comm. Partial Differential Equations**18**(1993), no. 9-10, 1631–1652. MR**1239926**, 10.1080/03605309308820988**19.**T. Tang and Zhen Huan Teng,*The sharpness of Kuznetsov’s 𝑂(√Δ𝑥)𝐿¹-error estimate for monotone difference schemes*, Math. Comp.**64**(1995), no. 210, 581–589. MR**1270625**, 10.1090/S0025-5718-1995-1270625-9**20.**Z.-H. Teng and P. W. Zhang,*Optimal -rate of convergence for viscosity method and monotone scheme to piecewise constant solutions with shocks*, 1994. To appear in SIAM J. Numer. Anal.

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Additional Information

**Tao Tang**

Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Email:
ttang@sfu.ca

**Zhen-huan Teng**

Affiliation:
Department of Mathematics, Peking University, Beijing 100871, China

Email:
tengzh@sxx0.math.pku.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-97-00822-3

Keywords:
Hyperbolic conservation laws,
error estimate,
viscosity methods,
piecewise smooth

Received by editor(s):
November 2, 1995

Received by editor(s) in revised form:
April 5, 1996

Additional Notes:
Research of the first author was supported by NSERC Canada Grant OGP0105545.

Research of the second author was supported by the National Natural Science Foundation of China and the Science Fund of the Education Commission of China.

Article copyright:
© Copyright 1997
American Mathematical Society