On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

TL;DR

This paper introduces a new, efficient method to verify the strong stability of one-step explicit upwind schemes for the 1D advection equation using the Kreiss-Lopatinskii theory and a novel determinant-based approach.

Contribution

A new intrinsic Kreiss-Lopatinskii determinant is proposed, enabling a robust and computationally inexpensive stability check for upwind schemes.

Findings

The method successfully assesses stability of the Beam-Warming scheme.

The approach simplifies stability analysis via winding number computation.

It provides a practical tool for boundary condition stability verification.

Abstract

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss-Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss-Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to elate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam-Warming scheme together with the simplified inverse Lax-Wendroff procedure at the boundary.