B-theory of general linear methods for Volterra functional differential equations

Abstract

B-theory of general linear methods (GLMs) for nonlinear Volterra functional differential equations (VFDEs) is established, which provides unified theoretical foundation for the study of GLMs when applied to nonlinear initial value problems (IVPs) in stiff ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice, and can be regarded as extension of the B-theory of Runge–Kutta methods for VFDEs presented by the same author in a previous paper. The extension from Runge–Kutta methods to the much more general class of GLMs is of essential importance since there exist many B-stable methods which are not one-step Runge–Kutta methods but can be regarded as special cases of GLMs.