A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

Abstract

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N−2), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results.

Keywords

• singularly perturbed
• Volterra integro-differential equation
• uniformly convergent
• adaptive grid method