Exact and nonstandard numerical schemes for linear delay differential models

Abstract

Delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions. In this work, exact numerical schemes for a general linear delay differential model, as well as for the special case of a pure delay model, are obtained. Based on these exact schemes, a family of nonstandard methods, of increasing order of accuracy and simple computational properties, is proposed. Dynamic consistency of the new nonstandard methods are proved, and illustrated with numerical examples, for asymptotic stability, positive preserving properties, and oscillation behaviour.