High-order methods for the numerical solution of Volterra integro-differential equations

摘要

If a first-order Volterra integro-differential equation is solved by collocation in the space of continuous polynomial splines of degree m⩾1, with collocation occuring at the Gauss-Legendre points, then the resulting approximation u converges, at its knots, like O(h2m), while its derivative u′ exhibits only O(hm)-convergence. This paper deals with the question of how to choose the collocation points so that both u and u′ converge like O(hq∗), with q∗ maximal.