Collocation-based two-step Runge-Kutta methods

Abstract

In this paper we propose a new cla.ss of numerical methods of Runge-Kutta type for integrating initial-value problems for first-order ODEs, that is two-step Runge-Kutt~ (TRK) methods. The advantage of TRK methods is that for a given order of accuracy p we can construct methods of order p with any desired number of implicit relations. In the very first iavestigation, the TRK methods Bre shown to be efficient numerical integration methods.

Derivation and implementation of Two-Step Runge-Kutta pairs

“…the function f per step and hence are not as efficient as, for example, linear multistep methods, when the derivative evaluations are relatively expensive. To seek compromises between the strengths and weaknesses of the standard methods, a number of authors [4], [5], [7], [8], [9], [10], [12], [14], [15], [16], [17], [18], [19], [20], [23], [24] have studied the possibility of using approximations to the solution and its derivatives at two consecutive steps. This approach leads to the general class of two-step Runge-Kutta (TSRK) methods of the form Y' = ujyi_l + (1 -uj)yi f h^(ajkf(Yk 1) + bjkf(Yik )) , implementations exploit explicit RK pairs for efficiency, and so the focus here is on the derivation of TSRK pairs suitable for implementation with variable stepsizes.…”