Selection problems of Z² -periodic entropy solutions and viscosity solutions

Z²-periodic entropy solutions of hyperbolic scalar conservation laws and Z²-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate Z²-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural Z²-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.