More on Stochastic and Variational Approach to the Lax-Friedrichs Scheme

TL;DR

This paper extends the stochastic and variational analysis of the Lax-Friedrichs scheme to demonstrate its stability, long-term behavior, and error bounds for hyperbolic conservation laws, linking it to viscosity solutions and KAM theory.

Contribution

It introduces new stability, asymptotic, and error estimates for the Lax-Friedrichs scheme using variational methods and viscosity solutions, and applies these to finite difference approximations of KAM tori.

Findings

Proved time-global stability of the scheme.

Established large-time behavior and error estimates.

Applied to finite difference approximation of KAM tori.

Abstract

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.