A Convergent Finite Volume Method for a Kinetic Model for Interacting Species

TL;DR

This paper introduces a convergent upwind finite volume method for a coupled system of kinetic equations modeling interacting species, ensuring mass conservation, positivity, and stability, with numerical validation.

Contribution

It presents a novel finite volume scheme for coupled kinetic models that guarantees convergence, mass conservation, and positivity, addressing challenges in simulating multi-species interactions.

Findings

The scheme conserves mass and preserves positivity.

Convergence of the numerical method is demonstrated.

Numerical experiments validate the scheme's effectiveness.

Abstract

We propose an upwind finite volume method for a system of two kinetic equations in one dimension that are coupled through nonlocal interaction terms. These cross-interaction systems were recently obtained as the mean-field limit of a second-order system of ordinary differential equations for two interacting species. Models of this kind are encountered in a myriad of contexts, for instance, to describe large systems of indistinguishable agents such as cell colonies, flocks of birds, schools of fish, herds of sheep. The finite volume method we propose is constructed to conserve mass and preserve positivity. Moreover, convex functionals of the discrete solution are controlled, which we use to show the convergence of the scheme. Finally, we investigate the scheme numerically.