Collocation-based two-step Runge-Kutta methods

Abstract

In this paper we propose a new class of numerical methods of Runge-Kutta type for integrating initial-value problems for first-order ODEs, that is two-step Runge-Kutta (TRK) methods. The advantage of TRK methods is that for a given order of accuracyp we can construct methods of orderp with any desired number of implicit relations. In the very first investigation, the TRK methods are shown to be efficient numerical integration methods.