Stability of finite difference schemes for the hyperbolic initial boundary value problem by winding number computations

TL;DR

This paper introduces a numerical method using winding number computations to efficiently verify the strong stability of finite difference schemes for the advection equation, enhancing stability analysis tools.

Contribution

It presents a novel approach employing the intrinsic Kreiss-Lopatinskii determinant and winding number calculations to assess scheme stability.

Findings

The method effectively verifies stability of O3 and LW5 schemes.

Winding number approach is robust and computationally inexpensive.

Application to boundary reconstruction procedures demonstrated.

Abstract

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit finite difference schemes for the one-dimensional advection equation with an inflow boundary condition. The strong stability is studied using the Kreiss-Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss-Lopatinskii determinant, which possesses the same regularity as the vector bundle of discrete stable solutions. By applying standard results of complex analysis to this determinant, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the O3 scheme and the fifth-order Lax-Wendroff (LW5) scheme together with a reconstruction procedure at the boundary.