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.

翻译：本文针对一种表现出丰富复杂行为的非局部活性物质系统，开发并分析了一种有限体积格式。所研究的模型源自描述觅食蚁群与信息素动力学耦合的随机相互作用粒子系统。在本工作中，我们证明了当网格尺寸和时间步长趋于零时，唯一的数值解收敛于唯一的弱解。我们还给出了离散长时间估计，证明了某些范数在所有时间均得以保持，且与网格尺寸和时间步长无关。特别地，我们证明了高阶正则性估计，这为连续抛物型高阶正则性估计提供了离散类比。最后，我们数值研究了该格式的收敛速率，并提供了多重亚稳态稳态解存在的示例。