Finite Volume Methods for Two-Dimensional Nonlocal Partial Differential Equation with a Gradient Flow Structure

Abstract This paper delves into a class of nonlinear nonlocal partial differential equations, characterized by a gradient flow structure, as previously outlined in the literature. Finite volume methods are employed to investigate the numerical solutions of this model in two-dimensional settings. It reveals that the semi-discrete numerical scheme satisfies the entropy dissipation and the fully discrete numerical scheme preserves the positivity through a meticulous combination of numerical scheme definitions, upwind numerical fluxes and sophisticated estimation techniques. Furthermore, a novel contribution is made by demonstrating that the semi-discrete numerical scheme preserves the positivity and the fully discrete numerical scheme satisfies the entropy dissipation. These fundamental properties, pivotal in guaranteeing the convergence of numerical solutions towards steady states, are rigorously proven utilizing the invariant region method, complemented by detailed norm estimations and rigorous analytical techniques. Two-dimensional numerical experiments validate the entropy dissipation and the long-time convergence of numerical solutions generated by the scheme. The numerical analysis methodology and computational findings in this work offer valuable insights for research on models with gradient flow structures.