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

Boutin Benjamin (1) , Pierre Le Barbenchon (1) , Seguin Nicolas (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 IMAG - Institut Montpelliérain Alexander Grothendieck

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.

Mots clés: boundary conditions; Kreiss-Lopatinskii determinant; GKS-stability; finite-difference methods; inverse Lax-Wendroff

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Boutin Benjamin, Pierre Le Barbenchon, Seguin Nicolas. On the stability of totally upwind schemes for the hyperbolic initial boundary value problem. IMA Journal of Numerical Analysis, In press, ⟨10.1093/imanum/drad040⟩. ⟨hal-03732720v3⟩