A Global Convergence Theorem for a Class of Parallel ContinuousExplicit Runge - Kutta Methods and Vanishing Lag DelayDi erential

Iterated continuous extensions (ICEs) are continuous explicit Runge-Kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay diierential equations (DDEs). ICEs have a particular r^ ole in the explicit solution of DDEs with vanishing lags. They may be regarded as parallel continuous explicit Runge-Kutta (PCERK) methods, as they allow advantage to be taken of parallel architectures. ICEs can also be related to a collocation method. The purpose of this paper is to provide a theorem giving the global order of convergence for variable-step implementations of ICEs applied to state-dependent DDEs with and without vanishing lags. Implications of the theory for the implementation of this class of methods are discussed and demonstrated. The results establish that our approach allows the construction of PCERK methods whose order of convergence is restricted only by the continuity of the solution.