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 one LU 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.

Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type

“…The idea of constructing RK methods which integrate exactly a set of linearly independent functions different from the polynomials has been proposed by several authors (see for example [4,[11][12][13]19,21]). This idea consists in selecting the available parameters of the method (5) and (6) in order to be exact for a set of linearly independent functions…”

“…Some results on the existence of a unique solutions for the coefficients of an EFRK method are obtained in [11,12], and several authors [3,5,7,11,13,19] have derived methods with variable coefficients that are able to integrate exactly first or second order differential systems whose solutions belong to the linear space generated by the set of functions {1, t, . .…”

“…The construction of Runge-Kutta (RK) methods for the numerical solution of ODEs which have periodic or oscillating solutions has been considered extensively in the literature (see [2][3][4][5][6][7][8][11][12][13][16][17][18][19][20][21][22] and the references therein) with the aim of using the available information on the solutions to derive more accurate and/or efficient algorithms than the general purpose algorithms for such types of problems. A detailed survey including an extensive bibliography on this subject can be found in [10].…”

On functionally-fitted Runge–Kutta methods

“…By construction, FRK methods are inherently implicit. But semi-implicit or diagonally-implicit methods can be obtained by imposing a lower triangular pattern to the RK matrix [10].…”

Block third derivative method based on trigonometric polynomials for periodic initial-value problems

“…(1) exponential fitting which take advantage of the special properties of the solution of (1) that may be known in advance have been proposed (see [4,5,13,13,14,21,[24][25][26]30,33]). It is also shown that the TFBTDM can be used to solve the general second-order IVPs of the form…”