Functionally fitted explicit pseudo two-step Runge–Kutta methods

Explicit pseudo two-step Runge–Kutta (EPTRK) methods belong to the wider class of general linear multistep methods. The particularity of EPTRK methods is that they do not use the last two iterates as conventional two-step methods do. Rather, they predict the intermediate stage values and combine them with the last iterate to obtain the next iterate. EPTRK methods were initially designed to suit parallel computers, but they have been shown to achieve arbitrary high-order and thus can be useful as conventional explicit RK methods on sequential computers as well. Our contribution in this paper is to present a new family of functionally fitted EPTRK methods aimed at integrating an equation exactly if its solution is a linear combination of a chosen set of basis functions. We use a variation of collocation techniques to show that this new family, which we call FEPTRK, shares the same accuracy properties as EPTRK. The added advantage is that FEPTRK can use specific fitting functions to capitalize on the special properties of the problem that may be known in advance.