One step integration methods with maximum stability regions

Abstract

One step integration methods, using K function evaluations, for the solution of a system of ordinary differential equations dv/ dt= A⋅ v are evaluated. A general expression for a class of methods is found for all positive integers K. This class of methods is proven to maximize the interval on the imaginary axis which is contained in the stability region such that the stability constraint is optimized. It is proven that every method with this optimal stability property has a polynomial M defined by y 1+, Δt = M⋅ v i for which M( iS l )= exp( iS l π/2) where S l = K− l.