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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 $O( \epsilon \vert \log \epsilon \vert + \epsilon )$ in the $L^1$-norm, which is an improvement of the $O( \sqrt {\epsilon })$ upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to $O(\epsilon )$.

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Additional Information

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

Zhen-huan Teng
Affiliation: Department of Mathematics, Peking University, Beijing 100871, China

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