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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
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