Convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods

We have recently developed a generic approach to solving retarded and neutral delay differential equations (DDEs). The approach is based on the use of an explicit continuous Runge-Kutta formula and employs defect control. The approach can be applied to problems with state-dependent delays and vanishing delays. In this paper we determine convergence properties for the numerical solution associated with methods that implement this approach. We first analyze such properties for retarded DDEs and then the analysis is extended to the case of neutral DDEs (NDEs). Such an extension is particularly important since NDEs have received little attention in the literature. The main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect. Our analysis can also be applied directly to other numerical methods (which use defect error control) based on Runge-Kutta formulas for DDEs.