A GLOBAL CONVERGENCE THEOREM FOR A CLASS OF PARALLEL CONTINUOUS EXPLICIT RUNGE-KUTTA METHODS AND VANISHING LAG DELAY-DIFFERENTIAL EQUATIONS

Iterated continuous extensions (ICEs) are continuous explicit Runge-Ku tta methods developed for the numerical solution of evolutionary probl ems in ordinary and delay differential equations (DDEs). ICEs have a p articular role in the explicit solution of DDEs with vanishing lags. T hey may be regarded as parallel continuous explicit Runge-Kutta (PCERK ) methods, as they allow one to take advantage of parallel architectur es. ICEs can also be related to a collocation method. The purpose of t his paper is to provide a theorem giving the global order of convergen ce for variable-step implementations of ICEs applied to state-dependen t DDEs with and without vanishing lags. Implications of the theory for the implementation of this class of methods are discussed and demonst rated. 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.