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}.

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