Error estimates of a theta-scheme for second-order mean field games

TL;DR

This paper introduces a new finite-difference theta-scheme for second-order mean field games, providing stability, monotonicity, and convergence rate analysis under certain regularity and CFL conditions.

Contribution

The paper develops a novel theta-method-based finite-difference scheme for second-order mean field games, with proven stability, monotonicity, and an explicit convergence rate estimate.

Findings

Proven stability and monotonicity under CFL condition

Established convergence rate of order O(h^r)

Validated scheme's effectiveness for regular solutions

Abstract

We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our theta-scheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order $\mathcal{O}(h^r)$ for the theta-scheme, where $h$ is the step length of the space variable and $r \in (0,1)$ is related to the H\"older continuity of the solution of the continuous problem and some of its derivatives.