Stability with large step sizes for multistep discretizations of stiff ordinary differential equations

Abstract:

In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter $\varepsilon$ characterizes the stiffness of these systems. We approximate a system of ODE’s in this set by a general class of multistep discretizations which includes both one-leg and linear multistep methods. We determine sufficient conditions under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of ODE’s, when the step size resolves the slow time scale but not the fast one. We call this property stability with large step sizes. The theory presented in this paper lets us compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff ODE’s. In particular, we show that one-leg methods have better stability properties with large step sizes than their linear multistep counterparts. This observation is consistent with results obtained by Dahlquist and Lindberg [11], Nevanlinna and Liniger [33] and van Veldhuizen [42]. Our theory also allows us to relate the concept of D-stability (van Veldhuizen [42]) to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes.