On the computation of the effective Hamiltonian in the non convex case

In this paper we propose a method to compute the effective Hamiltonian, a classical problem arising e.g. in weak KAM theory and homogenization. We will focus our attention on the case of non convex Hamiltonians related to differential games where the effective Hamiltonian gives information regarding the ergodicity of the game. The method is based on solution of the Hamilton—Jacobi—Isaacs equation and gives an approximation of the effective Hamiltonian via a coupling between a dynamic programming scheme for pursuit-evasion games and the techniques adapted to solve the cell problem in the convex case. Some tests will be presented in the last section.