On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions

TL;DR

This paper establishes a convergence rate for upwind finite difference methods applied to strongly degenerate convection-diffusion equations in multiple dimensions, demonstrating robustness even as diffusion effects vanish.

Contribution

It provides the first explicit error estimate for these methods in multiple dimensions, using kinetic formulations to handle degeneracy and complex spatial behavior.

Findings

Error rate of O(Δx^{2/(19+d)}) for the numerical method

Error estimate remains valid as diffusion effects vanish

Method effectively handles degeneracy in convection-diffusion equations

Abstract

We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local $L^1$-error between the exact and numerical solutions is $\mathcal{O}(\Delta x^{2/(19+d)})$, where $d$ is the spatial dimension and $\Delta x$ is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation.