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

Abstract

Jauslin–Kreiss–Moser and W E made clear the connection between the Aubry–Mather theory and the inviscid forced Burgers equation with a -periodic forcing term and established the smooth approximation of -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 -periodic entropy solutions. This result leads to difference approximations of the corresponding effective Hamiltonian and -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.