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

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.