Instability of difference models for hyperbolic initial boundary value problems

Abstract

AbstractA th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0. To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundström. In particular we investigate l2‐instability with respect to both initial and boundary data and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with C = 0 versus C > 0.