# Analysis of the implicit upwind finite volume scheme with rough coefficients

## TL;DR

This paper analyzes the implicit upwind finite volume scheme for the linear continuity equation with low regularity coefficients, establishing a convergence rate of at least 1/2 in a weak sense.

## Contribution

It provides the first convergence rate analysis of the scheme in the DiPerna-Lions setting with rough coefficients and unstructured meshes.

## Key findings

- Convergence rate of at least 1/2 in logarithmic Kantorovich-Rubinstein distance.
- Error bounds applicable to low regularity velocity fields.
- Validation of the scheme's effectiveness in rough coefficient scenarios.

## Abstract

We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation 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 prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at least 1/2. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.