L1 Error Estimates for Difference Approximations Of Degenerate Convection-Diffusion Equations

TL;DR

This paper provides an analysis of monotone difference schemes for degenerate convection-diffusion equations, establishing an L1 error bound proportional to the cube root of the spatial grid size, thus advancing numerical approximation understanding.

Contribution

It introduces a novel L1 error estimate for difference schemes applied to degenerate convection-diffusion equations, highlighting the scheme's convergence rate.

Findings

L1 error between approximation and entropy solution is bounded by a constant times the cube root of grid size.

The analysis applies to strongly degenerate convection-diffusion equations in one dimension.

The results improve understanding of numerical scheme accuracy for nonlinear degenerate PDEs.

Abstract

We analyze monotone difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1 difference between the approximate solutions and the unique entropy solution is bounded above by a constant times the cube root of the spatial discretization parameter.