Stability of the method of lines

Abstract

It is well known that a necessary condition for the Lax-stability of the method of lines is that the eigenvalues of the spatial discretization operator, scaled by the time stepk, lie within a distanceO(k) of the stability region of the time integration formula ask→0. In this paper we show that a necessary and sufficient condition for stability, except for an algebraic factor, is that the ε-pseudo-eigenvalues of the same operator lie within a distanceO(ε)+O(k) of the stability region ask, ε→0. Our results generalize those of an earlier paper by considering: (a) Runge-Kutta and other one-step formulas, (b) implicit as well as explicit linear multistep formulas, (c) weighted norms, (d) algebraic stability, (e) finite and infinite time intervals, and (f) stability regions with cusps. In summary, the theory presented in this paper amounts to a transplantation of the Kreiss matrix theorem from the unit disk (for simple power iterations) to an arbitrary stability region (for method of lines calculations).