An improve error estimate for the finite difference approximation to degenerate convection-diffusion equations

Abstract

We consider first-order finite difference schemes for a nonlinear degenerate convection-diffusion equation in one space dimension, and prove an $L^1_{loc}$ error estimate. Precisely, we prove that the $L^1_{loc}$ difference between the approximate solution given by the semidicrete scheme and the unique entropy solution converges at a rate $\mathcal{O}(\Delta x^{1/7})$ where $\Delta x$ is the spatial mesh size. We then prove a similar result for the fully discrete implicit scheme. We also prove all the needed properties regarding the semidiscrete scheme. The numerical flux function $F$ is assumed to be both monotone and split in the following sense: \begin{displaymath} F(u,v) = F_1(u) + F_2(v) \end{displaymath} for some $F_1$ and $F_2$ in $C^1(\mathbb{R})$.