Combined functional continuous method for delay differential equations

In the paper a combined numerical method for discrete delay differential equations is presented. The method is an embedded pair of type two explicit Runge—Kutta methods of order four: a continuous method with six stages and a stage-continuous method with seven stages. Their combination provides an effective solution of discrete delay differential equations. The combined method remains explicit for any values of the delay: for small values the stage-continuous scheme is used while for large delays a faster continuous scheme is applied. The scheme to use is chosen automatically based on whether the delay falls into the current step and a switch to the stage-continuous scheme can be made at any stage when required. The embedding of the methods lets to minimize the required number of the right-hand side function computations. The order conditions and the proof of their resolvability with the stated number of stages are presented. Tests, confirming the effectiveness of the proposed methods, are made.