×

zbMATH — the first resource for mathematics

Viscosity methods for piecewise smooth solutions to scalar conservation laws. (English) Zbl 0864.65060
Summary: 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 the viscosity solution to the inviscid solution is bounded by \(O(\varepsilon |\log \varepsilon |+ \varepsilon)\) in the \(L^1\)-norm, which is an improvement of the \(O(\sqrt \varepsilon)\) upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to \(O(\varepsilon)\).

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. S. Bahvalov, Error estimates for numerical integration of quasilinear first-order equations, Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961), 771 – 783 (Russian).
[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. · Zbl 0377.35051 · doi:10.1512/iumj.1977.26.26088 · doi.org
[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. · Zbl 0792.35115 · doi:10.1007/BF00410614 · doi.org
[5] Eduard Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527 – 536. · Zbl 0645.65052
[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. · Zbl 0343.76023 · doi:10.1002/cpa.3160300506 · doi.org
[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. · Zbl 0351.76070 · doi:10.1002/cpa.3160290305 · doi.org
[8] Gray Jennings, Discrete shocks, Comm. Pure Appl. Math. 27 (1974), 25 – 37. · Zbl 0304.65063 · doi:10.1002/cpa.3160270103 · doi.org
[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. · Zbl 0689.35001
[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). · Zbl 0354.35021
[11] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537 – 566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 · doi.org
[12] Jian-Guo Liu and Zhou Ping Xin, \?\textonesuperior -stability of stationary discrete shocks, Math. Comp. 60 (1993), no. 201, 233 – 244. · Zbl 0795.65059
[13] Randall J. LeVeque, Numerical methods for conservation laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. · Zbl 0847.65053
[14] Bradley J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), no. 6, 1074 – 1081. · Zbl 0584.65059 · doi:10.1137/0722064 · doi.org
[15] O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95 – 172. · Zbl 0131.31803
[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. · Zbl 0508.35002
[17] David G. Schaeffer, A regularity theorem for conservation laws, Advances in Math. 11 (1973), 368 – 386. · Zbl 0267.35009 · doi:10.1016/0001-8708(73)90018-2 · doi.org
[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. · Zbl 0807.35091 · doi:10.1080/03605309308820988 · doi.org
[19] T. Tang and Zhen Huan Teng, The sharpness of Kuznetsov’s \?(\sqrt \Delta \?)\?\textonesuperior -error estimate for monotone difference schemes, Math. Comp. 64 (1995), no. 210, 581 – 589. · Zbl 0845.65053
[20] Z.-H. Teng and P. W. Zhang, Optimal \(L^1\)-rate of convergence for viscosity method and monotone scheme to piecewise constant solutions with shocks, 1994. To appear in SIAM J. Numer. Anal.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.