Superconvergence for multistep collocation

Abstract

One-step collocation methods are known to be a subclass of implicit Runge-Kutta methods. Further, one-leg methods are special multistep one-point collocation methods. In this paper we extend both of these collocation ideas to multistep collocation methods with k previous meshpoints and m collocation points. By construction, the order is at least m + k − 1 m + k - 1 . However, by choosing the collocation points in the right way, order 2 m + k − 1 2m + k - 1 is obtained as the maximum. There are ( m + k − 1 k − 1 ) \left ( {\begin {array}{*{20}{c}} {m + k - 1} \\ {k - 1} \\ \end {array} } \right ) sets of such "multistep Gaussian" collocation points.