A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

Butcher (1) derived the order conditions of Runge-Kutta method using the theory of trees. Franco (2) constructed the embedded EFRKN4(3) methods relying on FSAL technique. Van de Vyver (3) constructed embedded phase-fitted modified Runge-Kutta method of order five and four based on the FSAL technique for solving the radial Schrödinger equation. Fang et al. (4) developed a new fifth-order Runge-Kutta method and embedded RK5(4) pair based on FSAL technique adapted for solving oscillatory problems. Chan and Tsai (5) constructed an explicit Two Derivative Runge-Kutta (TDRK) methods of order up to seven that include one function evaluation of and a minimal number of function evaluations of . They derived the order conditions of TDRK method based on Butcher's theory of trees in (1). Fang et al. (6) proposed fourth-order extended Runge-Kutta Nystrom (EKRN) methods and then they derived embedded EKRN4(3) pairs based on FSAL property to solve second-order ODEs with perturbed oscillators solutions. Very recently, Ahmad and Senu (7) have proposed a new explicit TDRK method of order four based on the FSAL property for solving first order ODEs.
Here in this paper, motivated by Chan and Tsai (5) and Ahmad and Senu (7), we developed a new four-stage fifth-order TDRK method designed utilizing the FSAL technique. The preliminaries of TDRK methods are presented in Section 2. In Section 3, a new TDRK method with FSAL property is derived. In Section 4, we analyzed the stability of the proposed method. In Section 5, numerical tests are given to demonstrate the efficiency of our TDRK method when it is compared with other Runge-Kutta methods in the scientific literature. Finally, in Section 6, we give some conclusion.

Preliminaries:
In this work, we are interested in the efficient numerical method for solving first-order ordinary differential equations (ODEs) (1). We consider the special explicit TDRK methods studied in (5) According to Chan and Tsai (5), the order conditions for new TDRK method up to five are presented as follows order 2: order 3: order 4: order 5: In practice, the following Nystrom row assumption is helpful in (8) A Fifth Order TDRK with FSAL Property: A new TDRK method with "First Same As Last" (FSAL) property will be derived where = , = 1, … , − 1, and = 0.
The feature of "First Same As Last" (FSAL) technique is that the fourth stage can be reused as the first stage of the next step. Therefore, the efficient number of function evaluations is three per step. According to the Nystrom row assumption (7), we have In order to construct four-stage fifth-order TDRK method by using FSAL technique, solving the order conditions (3)-(6) simultaneously results in a solution with one free parameter 3 as follows; Choosing 3 = 4 5 , yields a fifth order TDRK method with FSAL property denoted as TDRK5F, which is given in the following Butcher tableau;
In Figure 1, the stability regions of the TDRK5F method is plotted. In this section, some test problems is solved using Matlab to show the efficiency of the new TDRK5F method as compared with some efficient RK methods which are selected from the scientific literature. The following methods are used in comparison:  TDRK5F: The new four-stages fifth-order TDRK method with FSAL property constructed in this paper.  RK5W: The five-stages fifth-order RK method given in (9).  RK5N: The six-stages fifth-order RK method given in (10).  RK5B: The six-stages fifth-order RK method given in (8).
The problems are integrated in the interval [0,10]. The accuracy criteria calculated by taking 10 of the maximum absolute error as follows: The accuracy = log 10 (max (| ( ) − |)).
The numerical results and the efficiency curves of the methods are presented in Tables 2,  3

Conclusion:
A new explicit two derivative Runge-Kutta method of order five with FSAL property is developed in this paper. Also, the linear stability of the new method is analyzed. From numerical results, we conclude that the new TDRK5F method is more efficient compared with the existing RK methods of the same order in the literature in terms of the number of function evaluations and the accuracy per step. The computations were implemented on a DELL PC with i3-3227U CPU, 4.0GB memory.