Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid (Journal of Computational Mathematics, 2022)
In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler fordifferential part and the composite numerical quadrature formula for integral part for whichboth an a priori and an a posteriori error analysis in the maximum norm are derived. Basedon the a priori error bound and mesh equidistribution principle, we prove that there existsa mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor functionand design a corresponding adaptive grid generation algorithm. Furthermore, we extendour presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate theeffectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delaydifferential equations with a turning point.
Keywords: Delay Volterra integro-differential equation; Singularly perturbed; Error analysis; Monitor function.