A Study of Strong and Weak Stability in Discretization Algorithms

Abstract

In numerical analysis the term is almost unexceptionally used to characterize the sensitivity (or rather insensitivity) of the result of an algorithm with respect to perturbations. In the numerical solution of dif? ferential equations by discretization methods an algorithm has been called stable if the solutions of the associated linear algorithm (the equa? tion) are bounded in terms of the inhomogeneities and the initial and/or boundary data and if this bound is uniform with respect to the discretiza? tion parameter h. This concept of stability makes no reference to the original infinitesimal problem which is to be approximated. Stability in this sense implies convergence (for h ?> 0) under quite general assumptions. But for the applicability of an algorithm convergence with h ?> 0 is not sufficient. Since the total computational effort is almost always proportional to a negative power of h and since perturbations other than local discretiza? tion errors can never be avoided, the algorithm should produce a good ap? proximation to the solution of the infinitesimal problem for reasonably large values of h and under additional perturbations (like round-off). For this purpose it is not only necessary that the local discretization errors be small but that the global effect of any local error be not unduly large. Thus it is natural to strengthen the stability requirement by asking that the al? gorithm be no more sensitive to local perturbations than lies in the nature of the original problem. If a stable algorithm has this additional feature it will be called a strongly stable discretization of the infinitesimal problem, otherwise a weakly stable one.1 In ?1 of this paper we will formalize this concept of strong and weak stability and derive a general criterion for strong stability. Subsequently, in ?2, we will apply our considerations to the well-known multistep methods for the numerical solution of systems of first order ordinary differential equations. There our more general concepts will practically coincide with what has been called strong stability and weak (in)stability so far. We will