The Green's function of the Lax-Wendroff and Beam-Warming schemes

TL;DR

This paper derives precise bounds for the Green's function of Lax-Wendroff and Beam-Warming schemes, clarifying their stability characteristics and providing uniform bounds for initial data of bounded variation.

Contribution

It establishes sharp Gaussian bounds for these schemes' Green's functions, revealing the spatial regions responsible for their instability and extending bounds to initial data of bounded variation.

Findings

Sharp Gaussian bounds for Green's functions are proven.

The spatial regions causing instability are identified.

Uniform bounds are extended to initial data of bounded variation.

Abstract

We prove a sharp uniform generalized Gaussian bound for the Green's function of the Lax-Wendroff and Beam-Warming schemes. Our bound highlights the spatial region that leads to the well-known (rather weak) instability of these schemes in the maximum norm. We also recover uniform bounds in the maximum norm when these schemes are applied to initial data of bounded variation.