In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

中文翻译：

---

具有交叉扩散的相互作用物种系统的有限体积方案的收敛

在这项工作中，我们提出了具有交叉扩散的两个非局部偏微分方程的耦合系统的正性保持半离散有限体积方案的收敛性。证明收敛结果的关键是建立正性以获得离散能量估计以获得紧凑性。我们在数值上观察到参考解的收敛性，在空间中具有一阶精度。此外，尽管存在自扩散的正则化效应，我们还是恢复了分离的静止状态。然而，如果自扩散或交叉扩散足够强，则会发生混合，同时两种密度保持连续。