Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations

摘要

This paper deals with the stability analysis of one-step methods for the numerical solution of initial value problems. Both stiff ordinary and partial differential equations are included.The problem is considered how to restrict the stepsize in the methods in order that they behave stable. We review and extend some recent results on this problem that are based on the use of stability regions in the complex plane.We focus on differential equations that are essentially more general than the classical test equation U′ (t) = λ·U(t) (with λ a complex constant). Further, the emphasis is on stability of the methods with respect to the maximum norm.