On the use of stability regions in the numerical analysis of initial value problems

Abstract

This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations.Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size.These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration.The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product).The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.