1School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China
2Department of Applied Mathematics, Nanjing Agricultural University, Nanjing 210095, China
3School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Copyright © 2016 Yanping Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.
In this paper, we are interested in the numerical integration of initial-value problems (IVPs) of first-order differential equations in the form whose solutions exhibit a pronounced exponential behaviour [1, 2]. For the numerical solution of problem (1), classical general-purpose methods such as Runge-Kutta (RK) methods and linear multistep methods (LMMs) can not produce satisfactory results due to the special structure of the problems. The possible ways to construct the numerical methods adapted to the character of the solutions can be obtained by using exponential fitting (EF) technique [3–9]. There have been a large number of results concerning this topic in [10–15].
In standard approach to deriving exponentially fitted Runge-Kutta(-Nyström) methods, the effect of the error in the internal stages to the error of the final stage is completely neglected. D’Ambrosio et al. revisited the EFRK(N) method with two stages by considering the error in the internal stages in [1, 2]. Ixaru  introduced A-stable explicit fourth-order Runge-Kutta methods with three stages by evaluating the errors in each internal stage.
For more accurate request, inspired by the work of Chan and Tsai , we improve the EFRK methods of D’Ambrosio et al.  by introducing the second derivative into the scheme. The paper is organized as follows. In Section 2, we construct two versions of exponentially fitted TDRK methods. The second version, revised version, has coefficients depending on the equation to be solved. In Section 3, we present the local truncations errors and analyze the linear stability of the new EFTDRK methods. Numerical experiments are reported in Section 4. Finally, in Section 5, we give some conclusive remarks.
2. Construction of the New Methods
For the numerical integration of (1), we consider the explicit TDRK methods of the form where . Method (2) can be expressed briefly by the Butcher tableau:
As scheme (2) indicates, at each step, this special explicit TDRK method involves only one evaluation of the function and evaluations of the function . The order conditions for the general-purpose TDRK method (2) are given by Chan and Tsai . In this section, we will consider the explicit two-stage TDRK method of the form In order to derive the exponentially fitted version of (4), we require that the internal stages exactly integrate the linear combination of the functions and the final stage exactly integrates functions For the propose of constructing new methods, we introduce the operator With the local assumption that there is no error in , we require that the linear operator (7) related to method (4) vanishes for the reference set (5) and obtain Solving the last equation, we have where As approaches zero, has the following Taylor series expansion: When is chosen, the second internal stage in (4) is exact for the linear combination of the functions in (5). Note that these three functions are solutions of the differential equation: Then the leading term of the error in the computation of the second stage can be given by