Analysis of the implicit upwind finite volume scheme with rough coefficients
This work provides a rigorous convergence analysis for numerical schemes in low-regularity settings, which is important for applications involving rough velocity fields.
The paper proves that the implicit upwind finite volume scheme for the linear continuity equation with rough coefficients converges at a rate of at least 1/2 in logarithmic Kantorovich-Rubinstein distances, providing a bound on weak convergence.
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.