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

TL;DR

This paper analyzes the stability of the two-dimensional Lax-Wendroff scheme with Neumann boundary conditions, providing optimal stability criteria and extending the analysis to complex boundary geometries like half-spaces and quarter-spaces.

Contribution

It offers a comprehensive stability analysis of the 2D Lax-Wendroff scheme with Neumann boundary conditions, including new extrapolation techniques at corners.

Findings

Energy method yields optimal stability criterion.

Neumann boundary conditions ensure scheme stability.

Proposed extrapolation at corners maintains stability.

Abstract

We study the stability of the two-dimensional Lax-Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the numerical level, we enforce the Neumann extrapolation boundary condition and show that the corresponding scheme is stable. Eventually we analyze the case of a quarter-space when the transport operator is outgoing with respect to both sides. We then enforce the Neumann extrapolation boundary condition on each side of the boundary and propose an extrapolation boundary condition at the numerical corner in order to maintain stability for the whole numerical scheme.