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

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.

arXiv comment27 pages. Revised version with improved main result uniform in the diffusion constant

References in corpus (7)

• Convex integration solutions to the transport equation with full dimensional concentration
• Lagrangian flows for vector fields with gradient given by a singular integral
• Probabilistic analysis of the upwind scheme for transport
• A quantitative theory for the continuity equation
• Convergence rates for upwind schemes with rough coefficients
• The Scharfetter--Gummel scheme for aggregation-diffusion equations
• Analysis of the implicit upwind finite volume scheme with rough coefficients