A high-order scheme for mean field games

In this paper we propose a high-order numerical scheme for time-dependent mean field games systems. The scheme, which is built by combining Lagrange-Galerkin and semi-Lagrangian techniques, is explicit, conservative, consistent, and stable for large time steps compared with the space steps. We provide a convergence analysis for the exactly integrated Lagrange-Galerkin scheme applied to the Fokker-Planck equation, and we propose an implementable version with inexact integration. Finally, we validate the convergence rate of the high order method proposed by numerical simulations of two Mean Field Games problems.