Convergence of a finite volume scheme for a model for ants
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909559503716352 |
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| author | Bruna, Maria Schmidtchen, Markus de Wit, Oscar |
| author_facet | Bruna, Maria Schmidtchen, Markus de Wit, Oscar |
| contents | We develop and analyse a finite volume scheme for a nonlocal active matter system known to exhibit a rich array of complex behaviours. The model under investigation was derived from a stochastic system of interacting particles describing a foraging ant colony coupled to pheromone dynamics. In this work, we prove that the unique numerical solution converges to the unique weak solution as the mesh size and the time step go to zero. We also show discrete long-time estimates, which prove that certain norms are preserved for all times, uniformly in the mesh size and time step. In particular, we prove higher regularity estimates which provide an analogue of continuum parabolic higher regularity estimates. Finally, we numerically study the rate of convergence of the scheme, and we provide examples of the existence of multiple metastable steady states. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24001 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of a finite volume scheme for a model for ants Bruna, Maria Schmidtchen, Markus de Wit, Oscar Numerical Analysis Analysis of PDEs 65M08, 65M12, 35B36 (Primary), 35K55, 35Q92, 92D50 (Secondary) We develop and analyse a finite volume scheme for a nonlocal active matter system known to exhibit a rich array of complex behaviours. The model under investigation was derived from a stochastic system of interacting particles describing a foraging ant colony coupled to pheromone dynamics. In this work, we prove that the unique numerical solution converges to the unique weak solution as the mesh size and the time step go to zero. We also show discrete long-time estimates, which prove that certain norms are preserved for all times, uniformly in the mesh size and time step. In particular, we prove higher regularity estimates which provide an analogue of continuum parabolic higher regularity estimates. Finally, we numerically study the rate of convergence of the scheme, and we provide examples of the existence of multiple metastable steady states. |
| title | Convergence of a finite volume scheme for a model for ants |
| topic | Numerical Analysis Analysis of PDEs 65M08, 65M12, 35B36 (Primary), 35K55, 35Q92, 92D50 (Secondary) |
| url | https://arxiv.org/abs/2503.24001 |