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 $m + k - 1$. However, by choosing the collocation points in the right way, order $2m + k - 1$ is obtained as the maximum. There are $\left ( {\begin {array}{*{20}{c}} {m + k - 1} \\ {k - 1} \\ \end {array} } \right )$ sets of such "multistep Gaussian" collocation points.