Superconvergence in Collocation and Implicit Runge-Kutta Methods for Volterra-Type Integral Equations of the Second Kind

The Numerical Solution of Nonlinear Volterra Integral Equations of the Second Kind by Collocation and Iterated Collocation Methods

“…These approximations will, in general, be different from the "exact" collocation approximation u defined by (1.1.6); due to the differing quadrature errors associated with (1.2.2a) and (1.2.2b) we have also # . However, the approximations u, t, and t all exhibit the same order of (global and local) convergence; in particular, we have the following results (see, e.g., Brunner and van der Houwen [10]; compare also de Hoog and Weiss [15], Brunner [7] only for (t,s)S but also for (t,s)S':={(t,s)'O<-_s<-_t+8}IxI, for some 3>0. This is, of course, a consequence of the special form of the quadrature approximation (1.2.2a).…”

On superconvergence techniques

“…Next, let us mention the works of Brunner [29][30][31][32][33][34], Brunner and Norsett [35] which are concerned with superconvergence when solving Volterra integral equations of the first and second kind by collocation methods. Some special equations have been studied by Goldberg, Lea and Miel [85] (airfoil equation), Larsen and Nelson [113] (discrete-ordinate equations in slab geometry).…”

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

“…(1.6) What can be said about the behaviour of collocation solutions u h that lie in smooth collocation spaces S (d) µ (I h ) with µ ≥ 2 and 1 ≤ d < µ, as h → 0? It was shown in 1967 by Loscalzo and Talbot [29] (compare also the related papers [28,26,27,21], and the survey papers by Schoenberg [41] and Nørsett [37]) that collocation in the "classical" spline space S (3) 4 (I h ) (which corresponds to µ = 4, d = 3) and with collocation at the points X h = {t n + c 1 h n :…”

“…, for any choice of distinct collocation parameters {c i } ( [2,3,9,8]). A number of additional convergence (or -in the terminology of the authors -stability) results in certain smoother piecewise polynomial spaces were recently established by Oja [38,39] and Oja and Saveljeva [40].…”