A high-order scheme for mean field games

TL;DR

This paper introduces a high-order numerical scheme combining Lagrange-Galerkin and semi-Lagrangian methods for solving time-dependent mean field games systems, demonstrating stability and convergence through analysis and numerical validation.

Contribution

It presents a novel high-order scheme for mean field games that is both consistent and stable for large time steps, along with convergence analysis and practical implementation details.

Findings

The scheme is stable for large time steps compared to space steps.

Convergence analysis confirms the scheme's accuracy for the Fokker-Planck equation.

Numerical experiments validate the convergence rate of the proposed method.

Abstract

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 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 proposed scheme through the numerical approximation of two mean field games systems.