Convergence Rate of Approximate Solutions to Conservation Laws with Initial Rarefactions

Abstract

The authors address the question of local convergence rate of conservative ${\textit{Lip}}^ + $-stable approximations $u^\varepsilon (x,t)$ to the entropy solution $u(x,t)$ of a genuinely nonlinear conservation law. This question has been answered in the case of rarefaction free, i.e., ${\textit{Lip}}^ + $-bounded, initial data. It has been shown that by post-processing $u^\varepsilon $, pointwise values of u and its derivatives may be recovered with an error as close to $O(\varepsilon )$ as desired, where $\varepsilon $ measures, in $W^{ - 1,1} $, the truncation error of the approximate solution $u^\varepsilon $.This paper extends the previous results by including ${\textit{Lip}}^ + $-unbounded initial data. Specifically, it is shown that for arbitrary $L_\infty \cap BV$ initial data, u and its derivatives may be recovered with an almost optimal, modulo a spurious log factor, error of $O(\varepsilon |\ln \varepsilon |)$. This analysis relies on obtaining new $Lip^ + $-stability estimates for the speed $a...