Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients

TL;DR

This paper analyzes the convergence rate of an implicit upwind finite volume scheme for advection-diffusion equations with low-regularity velocity fields, providing error estimates in weak convergence metrics.

Contribution

It establishes a convergence rate of at least one for the scheme in the DiPerna-Lions setting with rough coefficients, using logarithmic Kantorovich-Rubinstein distances.

Findings

Convergence rate of at least one on unstructured meshes.

Error estimates in terms of logarithmic Kantorovich-Rubinstein distances.

Provides bounds on weak convergence of numerical solutions.

Abstract

We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.