The construction of reducible quadrature methods for Volterra integral and integro-differential equations

A formal relationship between quadrature rules and linear multistep methods for ordinary differential equations is exploited for the generation of quadrature weights. Employing the quadrature rules constructed in this way, step-by-step methods for second kind Volterra integral equations and integro-differential equations are defined and convergence and stability results are presented. The construction of the quadrature rules generated by the backward differentiation formulae is discussed in detail. The use of these rules for the solution of Volterra type equations is proposed and their good performance is demonstrated by numerical experiments.