D-convergence and conditional GDN-stability of exponential Runge–Kutta methods for semilinear delay differential equations

Abstract

This paper is concerned with exponential Runge–Kutta methods with Lagrangian interpolation (ERKLMs) for semilinear delay differential equations (DDEs). Concepts of exponential algebraic stability and conditional GDN-stability are introduced. D-convergence and conditional GDN-stability of ERKLMs for semilinear DDEs are investigated. It is shown that exponentially algebraically stable and diagonally stable ERKLMs with stage order p, together with a Lagrangian interpolation of order q (q ≥ p), are D-convergent of order p. It is also shown that exponentially algebraically stable and diagonally stable ERKLMs are conditionally GDN-stable. Some examples of exponentially algebraically stable and diagonally stable ERKLMs of stage order one and two are given, and numerical experiments are presented to illustrate the theoretical results.