Stability analysis of explicit exponential integrators for delay differential equations

Abstract

The aim of this paper is to analyze stability properties of explicit exponential integrators for three kinds of delay differential equations. First, linear autonomous delay differential equations are studied, and sufficient conditions for P- and GP-contractivity of explicit exponential Runge–Kutta methods are given. Second, for linear nonautonomous test equations, PN- and GPN-stability of a particular Magnus integrator is investigated. It is shown that the Magnus integrator is GPN-stable and convergent of order two. Finally, for semilinear delay differential equations, RN- and GRN-stability of explicit exponential Runge–Kutta methods is studied and sufficient conditions for GRN-stability are derived. Some examples of P-, GP-, RN- and GRN-stable exponential integrators are given, and numerical experiments that illustrate the theoretical results are included.