| Issue |
ESAIM: M2AN
Volume 59, Number 6, November-December 2025
|
|
|---|---|---|
| Page(s) | 3301 - 3340 | |
| DOI | https://doi.org/10.1051/m2an/2025092 | |
| Published online | 17 December 2025 | |
Convergence of a finite volume scheme for a model for ants
1
Mathematical Institute, University of Oxford, Oxford, UK
2
Institute of Scientific Computing, Faculty of Mathematics, TU Dresden, Germany
3
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK
* Corresponding author: odewit8@gmail.com
Received:
31
March
2025
Accepted:
31
October
2025
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.
Mathematics Subject Classification: 35K55 / 35B36 / 65M08 / 65M12 / 35Q92 / 92D50
Key words: Active matter / emergent behaviour / finite volume methods / numerical analysis
© The authors. Published by EDP Sciences, SMAI 2025
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