The Neumann boundary condition for the two-dimensional Lax-Wendroff scheme. II

TL;DR

This paper investigates the stability of a two-dimensional Lax-Wendroff scheme with Neumann boundary conditions, proposing a modified energy method and boundary conditions to ensure stability and second order accuracy near corners.

Contribution

It introduces a modified energy approach and second order extrapolation boundary conditions for the 2D Lax-Wendroff scheme to achieve stability and accuracy.

Findings

Modified energy method ensures stability near boundaries

Second order extrapolation boundary conditions maintain accuracy

Scheme remains stable and accurate at corners

Abstract

We study the stability of a two-dimensional Lax-Wendroff scheme in a quarter-plane. Following our previous work, we aim here at adapting the energy method in order to study second order extrapolation boundary conditions. We first show on the one-dimensional problem why modifying the energy is a necessity in order to obtain stability estimates. We then study the two-dimensional case and propose a modified energy as well as second order extrapolation boundary and corner conditions in order to maintain second order accuracy and stability of the whole scheme, including near the corner.