Linear multistep methods for Volterra integral and integro-differential equations

Abstract

A general class of linear multistep methods is presented for numerically solving first- and second-kind Volterra integral equations, and Volterra integro-differential equations. These so-called VLM methods, which include the well-known direct quadrature methods, allow for a unified treatment of the problems of consistency and convergence, and have an analogue in linear multistep methods for ODEs, as treated in any textbook on computational methods in ordinary differential equations. General consistency and convergence results are presented (and proved in an Appendix), together with results of numerical experiments which support the theory.