  Z^2-periodic entropy solutions of hyperbolic scalar conservation laws and
Z^2-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. Ugo Bessi ('03) investigated the convergence
of approximate Z^2-periodic solutions to an exact one in the process of the
vanishing viscosity method, and characterized this physically natural
Z^2-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 exhibits
difference in characteristics between the two approximation techniques.
