A generalized Scharfetter-Gummel scheme for nonlocal cross-diffusion systems

TL;DR

This paper introduces a generalized Scharfetter-Gummel finite-volume scheme for nonlocal cross-diffusion systems, ensuring positivity, mass conservation, and entropy preservation, with proven convergence and numerical validation.

Contribution

It develops a novel generalized Scharfetter-Gummel discretization for nonlocal fluxes, addressing degeneracy issues and proving convergence to the continuous solution.

Findings

Scheme preserves positivity, mass, and entropy.

Convergence of the discrete solution to the continuous problem.

Numerical simulations demonstrate scheme effectiveness in 1D and 2D.

Abstract

An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the multidimensional torus is analyzed. The equations describe the dynamics of population species with repulsive or attractive interactions. The numerical scheme is based on a generalized Scharfetter-Gummel discretization of the nonlocal flux term. For merely integrable kernel functions, the scheme preserves the positivity, total mass, and entropy structure. The existence of a discrete solution and its convergence to a solution to the continuous problem, as the mesh size tends to zero, are shown. A key difficulty is the degeneracy of the generalized Bernoulli function in the Scharfetter-Gummel approximation. This issue is overcome by proving a uniform estimate for the discrete Fisher information, which requires both the Boltzmann and Rao entropy inequalities. Numerical simulations illustrate the features of the scheme in one and two space dimensions.