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

Abstract 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 (h 2m ) , while its derivative u ′ exhibits only O (h m ) -convergence. This paper deals with the question of how to choose the collocation points so that both u and u ′ converge like O (h q∗ ) , with q ∗ maximal.