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

TL;DR

This paper establishes an L1 error estimate for finite difference schemes approximating nonlinear degenerate convection-diffusion equations, showing convergence rates that depend on the diffusion type.

Contribution

It provides the first rigorous L1 error estimate for semi-discrete finite difference schemes solving degenerate convection-diffusion equations.

Findings

Error rate of O(Δx^{1/11}) for nonlinear degenerate cases

Error rate of O(Δx^{1/2}) for linear diffusion cases

Convergence rate independent of diffusion size in linear case

Abstract

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection-diffusion equations in one space dimension, and prove an L1 error estimate. Precisely, we show that the L1 loc difference between the approximate solution and the unique entropy solution converges at a rate O(\Deltax 1/11), where \Deltax is the spatial mesh size. If the diffusion is linear, we get the convergence rate O(\Deltax 1/2), the point being that the O is independent of the size of the diffusion