Difference approximation to Aubry-Mather sets of the forced Burgers equation

Jauslin-Kreiss-Moser andWE made clear the connection between the Aubry-Mather theory and the inviscid forced Burgers equation with a ℤ ²-periodic forcing term and established the smooth approximation of ℤ ²-periodic entropy solutions of the PDE. This paper presents results of a difference approximation to the Aubry-Mather sets. We prove the convergence of the Lax-Friedrichs scheme for the ℤ ²-periodic entropy solutions. This result leads to difference approximations of the corresponding effective Hamiltonian and ℤ ²-periodic viscosity solutions of the Hamilton-Jacobi equation. We numerically construct the Aubry-Mather sets through the approximate entropy solutions, based on the dynamical properties of the Aubry-Mather sets.