An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to ``translate'' the entropy calculations for the continuous equations to the finite-volume scheme, in particular to design discretizations of the mobilities, which guarantee a discrete chain rule even in the presence of nonlocal terms. Based on this idea, the existence of finite-volume solutions and the convergence of the scheme are proven. As a by-product, we deduce the existence of weak solutions to the continuous cross-diffusion system. Finally, we present some numerical experiments illustrating the behavior of the solutions to the nonlocal and associated local models.

翻译：提出并分析了一个适用于一维环面上非局部交叉扩散系统的隐式欧拉有限体积格式，该格式源于群体动力学。假设核函数满足细致平衡条件及弱交叉扩散条件，后者允许非对角系数为负值且核函数由指示函数定义。该格式保持了密度的非负性、质量守恒以及玻尔兹曼熵和拉奥熵的产生。关键思想是将连续方程的熵计算"平移"到有限体积格式中，特别是设计迁移率的离散化方法，即使在存在非局部项的情况下也能保证离散链式法则。基于这一思想，我们证明了有限体积解的存在性以及格式的收敛性。作为副产品，我们推导出了连续交叉扩散系统弱解的存在性。最后，我们通过数值实验展示了非局部模型及其关联局部模型解的行为。