New stability theorems concerning one-step numerical methods for ordinary differential equations

Abstract

A measure of the stability properties of numerical integration methods for ordinary differential equations is provided by their stability region, which is that region in the complex ( Δtλ) plane for which a given method is stable when applied to the differential equation dy dt=λy with a time-step Δ t. Free parameters which exist in numerical integration algorithms may be used to maximize, in some sense, the size of the stability region, rather than increasing the order of accuracy, as is usually done. We derive new results which set theoretical limits to this maximization process for one step, explicit methods. Specifically, if K is the number of function evaluation invoked, then: (i) we prove (Theorem 1) that if ϱ is the radius of the largest circle, tangent to the imaginary axis at the origin of the complex plane that is contained in the stability region S, then ϱ cannot exceed K. (ii) we also prove (Theorem 2) that the imaginary stability boundary S I (or maximum stable value of ∣ Δtλ∣ with λ imaginary) cannot exceed ( K − 1). While Theorem 1 is to our knowledge new, a limited form of theorem 2 ( K odd only) had been established in v.d. Houwen (1977). That the maximum imaginary boundary S I = ( K − 1) is attainable had been shown (constructively) for K odd. We show that this maximum is also reached for K = 2 and K = 4, and correct in the process an erroneous result in the above reference.