Reducible quadrature methods for Volterra integral equations of the first kind

Abstract

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.