On L1 convergence rate of viscous and numerical approximate solutions of genuinely nonlinear scalar conservation laws

We study the rate of convergence of the viscous and numerical approximate solution to the entropy solution of genuinely nonlinear scalar conservation laws with piecewise smooth initial data. We show that the O(epsilon\ log epsilon\) rate in L-1 is indeed optimal for viscous Burgers equation. Through the Hopf-Cole transformation, we can study the detailed structure of parallel to u(., t) u(epsilon)(., t)parallel to L-1. For centered rarefaction wave, the O(epsilon\ log epsilon\) error occurs on the edges where the inviscid solution has a corner, and persists as long as the j edges j remain. The O(epsilon\ log epsilon\) error must also occur at the critical time when a new shock forms automatically from the decreasing j part of the initial data; thus it is, in general, impossible to maintain O(epsilon) rate for all t >0. In contrast to the centered rarefaction wave case, the O(epsilon\ log epsilon\) error at critical time is transient. It resumes the O(epsilon) rate right after the critical time due to nonlinear effect. Similar examples of some monotone schemes, which admit a discrete version of the Hopf-Cole transformation, are also included.