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arXiv · 2601.01731

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

Ansgar Jüngel·Panchi Li·Zhiwei Sun
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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.

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Ansgar Jüngel, Panchi Li, Zhiwei Sun. 2026-01-05. A generalized Scharfetter-Gummel scheme for nonlocal cross-diffusion systems. https://arxiv.org/abs/2601.01731

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