  A stochastic and variational aspect of the Lax-Friedrichs scheme was applied
to hyperbolic scalar conservation laws by Soga [arXiv: 1205.2167v1]. The
results for the Lax-Friedrichs scheme are extended here to show its time-global
stability, the large-time behavior, and error estimates. The proofs essentially
rely on the calculus of variations in the Lax-Friedrichs scheme and on the
theory of viscosity solutions of Hamilton-Jacobi equations corresponding to the
hyperbolic scalar conservation laws. Also provided are basic facts that are
useful in the numerical analysis and simulation of the weak
Kolmogorov-Arnold-Moser (KAM) theory. As one application, a finite difference
approximation to KAM tori is rigorously treated.
