Viscosity methods for piecewise smooth solutions to scalar conservation laws

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 $O(\epsilon )$.