Convergence of a finite volume scheme for a model for ants

TL;DR

This work develops and analyzes a finite-volume scheme for a nonlinear, nonlocal Fokker–Planck model of ant dynamics coupled to a pheromone field. By proving convergence of the scheme to the unique weak solution and establishing robust long-time bounds, the study provides a rigorous computational framework for capturing metastable lane- and cluster-formation phenomena in active matter. The combination of discrete Morrey-type and Alikakos-type estimates, along with careful treatment of the nonlocal interaction $B[c]$, yields reliable, mass-preserving, and nonnegative approximations that illuminate the role of sensing distance and interaction variants in pattern formation. Numerically, the scheme demonstrates expected order of accuracy and reveals rich dynamics, including multiple metastable steady states and their eventual evolution to simpler structures depending on the interaction term.

Abstract

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.

Figures (6)

• Figure 1: Spatial density $\rho(t,x)$ from \ref{['eq:schemerho']} at times $t = 0.1$ (left) and $t = 1.0$ (right) for different mesh sizes for $D_T=10^{-1},\gamma=500,\mathrm{Pe}=2,\alpha=1,B=B_0,\Delta t=10^{-2}$. The initial condition (shown in grey on the left panel) is $f^0(x, \theta) = C \mathbf{1}_{|x\pm\frac{1}{8}|\leq\frac{1}{8}}$, with $C>0$ such that $\int f\mathrm{d} x\mathrm{d}\theta=1$.
• Figure 2: Evolution of $\rho(t,x)$ from \ref{['eq:schemerho']} and $p_2(t,x) = \int_0^{2\pi}\cos(2\theta)f(t,x,\theta)\mathrm{d}\theta$ for $B$-terms (a) $B_0$\ref{['eq:B0_sch']} and (b) $B_\lambda$\ref{['eq:blambdadisc']} with $\lambda=0.1$, for $D_T=10^{-1},\gamma=500,\mathrm{Pe}=2,\alpha=1,\Delta t=10^{-3},N=64$. The initial condition is $f^0(x, \theta) = C \mathbf{1}_{|x\pm\frac{1}{4}|\leq\frac{1}{8}}$, with $C>0$ such that $\int f^0\mathrm{d} x\mathrm{d}\theta=1$.
• Figure 3: Relative norm of the difference between numerical solutions for $B_\lambda$ and $B_\tau$, $\lambda=\tau$, with $D_T=10^{-1},\gamma=500,\mathrm{Pe}=2,N=64,\Delta t=10^{-2}, T=1.0$. The initial condition is $f^0(x, \theta) = C \mathbf{1}_{|x\pm\frac{1}{8}|\leq\frac{1}{8}}$, with $C>0$ such that $\int f\mathrm{d} x\mathrm{d}\theta=1$, as in \ref{['fig:rhoev']}.
• Figure 4: Relative error \ref{['eq:reler']} for numerical solutions to \ref{['eq:scheme']} with varying mesh size, for (a) $B_0$ interaction \ref{['eq:B0_sch']} and (b) $B_\lambda$\ref{['eq:blambdadisc']} with $\lambda=0.1$. The initial condition and the scheme are as in \ref{['fig:rhoev']}. Other parameters used: $T=1.0,D_T=10^{-1},\gamma=500,\mathrm{Pe}=2,\alpha=1,\Delta t=10^{-2}$.
• Figure 5: Nearest-neighbour interpolation for $B_\lambda$.

Theorems & Definitions (42)

• Definition 2.1: Weak solution
• Definition 2.2: Discretisation of the domain
• Definition 2.3: Definition of the scheme
• Definition 2.4: Piecewise constant interpolations
• Proposition 2.1: Existence, uniqueness and nonnegativity
• Theorem 2.1: Convergence of the scheme
• Lemma 3.1: Discrete Grönwall inequality
• Definition 3.1: Function space norms
• Lemma 3.2: Summation by parts
• proof