Nonlinear stability and D-convergence of Runge-Kutta methods for delay differential equations

Abstract

This paper deals with the stability and convergence of Runge-Kutta methods with the Lagrangian interpolation (RKLMs) for nonlinear delay differential equations (DDEs). Some new concepts, such as strong algebraic stability, GDN-stability and D-convergence, are introduced. We show that strong algebraic stability of a RKM for ODEs implies GDN-stability of the corresponding RKLM for DDEs, and that a strongly algebraically stable and diagonally stable RKM with order p, together with a Lagrangian interpolation of order q, leads a D-convergent RKLM of order min{ p, q + 1}.