A generalized Scharfetter-Gummel scheme for nonlocal cross-diffusion systems
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| Format: | Preprint |
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2026
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| _version_ | 1866912801831780352 |
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| author | Jüngel, Ansgar Li, Panchi Sun, Zhiwei |
| author_facet | Jüngel, Ansgar Li, Panchi Sun, Zhiwei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01731 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A generalized Scharfetter-Gummel scheme for nonlocal cross-diffusion systems Jüngel, Ansgar Li, Panchi Sun, Zhiwei Numerical Analysis 65M08, 65M12, 35Q92, 92D25 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. |
| title | A generalized Scharfetter-Gummel scheme for nonlocal cross-diffusion systems |
| topic | Numerical Analysis 65M08, 65M12, 35Q92, 92D25 |
| url | https://arxiv.org/abs/2601.01731 |