Error growth analysis via stability regions for discretizations of initial value problems

This paper deals with numerical methods for the solution of linear initial value problems. Two main theorems are presented on the stability of these methods. Both theorems give conditions guaranteeing a mild error growth, for one-step methods characterized by a rational function ϕ(z). The conditions are related to the stability regionS={z:z∈ℂ with |ϕ(z)|≤1}, and can be viewed as variants to the resolvent condition occurring in the reputed Kreiss matrix theorem. Stability estimates are presented in terms of the number of time stepsn and the dimensions of the space. The first theorem gives a stability estimate which implies that errors in the numerical process cannot grow faster than linearly withs orn. It improves previous results in the literature where various restrictions were imposed onS and ϕ(z), including ϕ′(z)≠0 forz∈σS andS be bounded. The new theorem is not subject to any of these restrictions. The second theorem gives a sharper stability result under additional assumptions regarding the differential equation. This result implies that errors cannot grow faster thann ^β, with fixed β<1. The theory is illustrated in the numerical solution of an initial-boundary value problem for a partial differential equation, where the error growth is measured in the maximum norm.