Convergence analysis of structure-preserving numerical methods for nonlinear Fokker-Planck equations with nonlocal interactions

A class of nonlinear Fokker-Planck equations with nonlocal interactions may include many important cases, such as porous medium equations with external potentials and aggregation-diffusion models. The trajectory equation of the Fokker-Plank equation can be derived based on an energetic variational approach. A structure-preserving numerical scheme that is mass conservative, energy stable, uniquely solvable, and positivity preserving at a theoretical level has also been designed in the previous work. Moreover, the numerical scheme is shown to satisfy the discrete energetic dissipation law and preserve steady states and has been observed to be second order accurate in space and first-order accurate time in various numerical experiments. In this work, we give the rigorous convergence analysis for the highly nonlinear numerical scheme. A careful higher order asymptotic expansion is needed to handle the highly nonlinear nature of the trajectory equation. In addition, two step error estimates (a rough estimate and a refined estimate) are necessary in the convergence proof. Different from a standard error estimate, the rough estimate is performed to control the nonlinear term. A few numerical results are also presented to verify the optimal convergence order and the preservation of equilibria.