A fully-discrete Semi-Lagrangian scheme for a first order mean field game problem

TL;DR

This paper introduces a fully-discrete Semi-Lagrangian numerical scheme for solving first order mean field game systems, providing existence, convergence proofs, and numerical validation.

Contribution

The work presents the first fully-discrete Semi-Lagrangian scheme for first order mean field games, with proven existence and convergence results in the scalar case.

Findings

Existence of solutions for the discretized scheme

Convergence of the scheme in the scalar case

Numerical simulations demonstrating scheme effectiveness

Abstract

In this work we propose a fully-discrete Semi-Lagrangian scheme for a {\it first order mean field game system}. We prove that the resulting discretization admits at least one solution and, in the scalar case, we prove a convergence result for the scheme. Numerical simulations and examples are also discussed.