A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method

Abstract

A special class of Runge-Kutta(-Nystrom) methods called functionally fitted (or functional fitting) Runge-Kutta (FRK) methods has recently been proposed by the author. This class of methods is designed to be exact, if the solution of the equation to be solved is an element of the linear space of given functions, which is called the basis functions. The purpose of this article is to develop a functionally fitted Runge-Kutta method that is cheap to implement. The method proposed in this paper is a three-stage explicit singly diagonally implicit Runge-Kutta (ESDIRK) method, which requires oneLU decomposition per step. This method is exact if the basis functions are properly chosen, and is moderately accurate even if the choice of the functions is inappropriate, since the method is shown to be of order 4 for general cases. An embedded pair of this type is also developed. Several numerical experiments show the superiority of the methods to conventional ones for the particular case that a suitable set of the basis functions can be found.