Convergence rates for upwind schemes with rough coefficients
Provides the first convergence rate analysis for upwind schemes with rough coefficients, addressing a gap in numerical analysis for transport problems with low-regularity data.
The paper proves that the explicit upwind finite volume scheme for continuity equations with rough velocity fields converges weakly at a rate of at least 1/2 in mesh size, measured in Kantorovich-Rubinstein distance, and shows this rate is optimal via a counterexample.
This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich-Rubinstein distance, which was recently used for stability estimates for the continuity equation by Seis [23]. Restricted to Cartesian meshes, our estimate shows that the rate of weak convergence is at least of order $1/2$ in the mesh size. The proof relies on a probabilistic interpretation of the upwind scheme Delarue and Lagoutière [9]. We complement the weak convergence result with an example that illustrates that for rough initial data no rates can be expected in strong norms. The same example suggests that the weak order $1/2$ rate is optimal.