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.

翻译：我们针对一维中通过非局部相互作用项耦合的两个动力学方程系统，提出了一种迎风有限体积法。这些交叉相互作用系统最近被作为两个相互作用物种的二阶常微分方程组平均场极限而获得。此类模型出现在众多情境中，例如描述大量不可区分个体组成的系统，如细胞群落、鸟群、鱼群和羊群。我们所提出的有限体积法被构建为守恒质量和保持正性。此外，离散解的凸泛函受到控制，我们利用这一点来证明该格式的收敛性。最后，我们对这个格式进行了数值研究。