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

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 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 Beam–Warming scheme together with the simplified inverse Lax–Wendroff procedure at the boundary.