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--Kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay differential equations (DDEs). ICEs have a particular role in the explicit solution of DDEs with vanishing lags. They may be regarded as parallel continuous explicit Runge--Kutta (PCERK) methods, as they allow one to take advantage 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.