Ergodic problems for Hamilton-Jacobi equations: yet another but efficient numerical method

TL;DR

This paper introduces an efficient Newton-like numerical method for solving ergodic problems in Hamilton-Jacobi equations, applicable to various complex scenarios including nonconvex Hamiltonians and mean field games, with demonstrated accuracy and speed.

Contribution

A novel Newton-like approach for ergodic Hamilton-Jacobi problems that handles diverse and complex cases more efficiently than existing methods.

Findings

Effective in solving cell problems for convex and nonconvex Hamiltonians

Applicable to weakly coupled systems and mean field games

Demonstrates high accuracy and reduced computational time in tests

Abstract

We propose a new approach to the numerical solution of ergodic problems arising in the homogenization of Hamilton-Jacobi (HJ) equations. It is based on a Newton-like method for solving inconsistent systems of nonlinear equations, coming from the discretization of the corresponding ergodic HJ equations. We show that our method is able to solve efficiently cell problems in very general contexts, e.g., for first and second order scalar convex and nonconvex Hamiltonians, weakly coupled systems, dislocation dynamics and mean field games, also in the case of more competing populations. A large collection of numerical tests in dimension one and two shows the performance of the proposed method, both in terms of accuracy and computational time.