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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics