Efficient two-derivative Runge-Kutta-Nyström methods for solving general second-order ordinary differential equations

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y'' (x) = f (x,y, y') including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods.


Introduction
In this article, we are interested in initial value problems (IVPs) of second-order ordinary differential equations (ODEs): () =  (,  () ,   ()) , where  ∈ R  ,  : R × R  × R  → R  are continuous vector valued functions.This type of problems arises naturally in many applied science fields such as the Kepler problems in celestial mechanics, quantum physics, and Newton's second law in classical mechanics (see Dormand [1], Hairer et al. [2], and Kristensson [3]).
Problems (1) in which the first derivative does not appear explicitly are an important subclass of second order (ODEs).Thus, several numerical methods for directly solving this subclass have been presented (see Dormand [1], Hairer et al. [2], Butcher [4], Lambert [5], and Senu [6]).In the case of direct solutions for the general second order (IVPs), some numerical methods have been proposed (see Chen et al. [7], Franco [8], Jator [9], Awoyemi [10], Wu et al. [11], Wu and Wang [12], and Chawla and Sharma [13]).The objective of this paper is to design STDRKN methods with a minimal number of function evaluation.This paper is organized as follows: In Section 3, we construct STDRKN methods; the stability analysis of STDRKN methods is discussed in Section 4; and numerical results are given in Section 5.
In this paper, a special part of TDRKN method is studied that has the form where An alternative expression of formula ( 5) is given as follows: where This STDRKN method can be written in Butcher's tableau as shown in Table 2.The STDRKN methods are explicit methods if  , = 0,  , = 0 for  ≤  and are implicit method if  , ̸ = 0,  , ̸ = 0 for  ≤ .STDRKN methods involve only one evaluation of  and many evaluations of  per step.

Construction of STDRKN Methods
In this section, our effort is to determine the coefficients of the STDRKN methods as given in (7).Hence, using the Taylor series expansion in (3) with the Taylor series expansion of   and    and by comparing the coefficients of the power of ℎ, we obtained the order conditions of STDRKN methods for  and   as in (10)-(18), while the rooted trees for STDRKN methods up to order five based on [7] are given in Table 3.
The following simplifying assumption is suggested in practice: The following are the order conditions for explicit STDRKN.

Stability of the STDRKN Methods
In this part, we study the linear stability of the STDRKN methods.We use the test problem (see [7,14]) Applying STDRKN method ( 5) to test problem (24), we obtain where with where The stability region for STDRKN4(2) method.
where V = ℎ and  = ℎ.The matrix (V, ) is called stability matrix, while the stability region of STDRKN method is defined by are eigenvalues of (V, ).The stability regions for STDRKN4(2) and STDRKN5(3) are shown in Figures 1 and  2, respectively.

Numerical Experiments
In this section, we test the effectiveness of the new methods of orders four and five on the same problems for comparison.The numerical methods used for comparison are given as follows: (i) STDRKN5( 3): new special explicit two-derivative RKN method of fifth order derived in this paper (ii) STDRKN4( 2): new special explicit two-derivative RKN method of fourth order derived in this paper (iii) TDRKN5(3): three-stage fifth-order two-derivative RKN method derived in [7] (iv) TDRKN4( 2): two-stage fourth-order two-derivative RKN method derived in [7] (v) RKNG5( 6): the classical six-stage fifth-order RKN method which is the limit method of ARKNGV5 as the frequency matrix  → 0 derived in [8] (vi) RKNG4: the classical four-stage fourth-order RKN method derived in [2].

Conclusion
In this study, the special class of explicit two-derivative Runge-Kutta-Nyström methods of order up to five that involve one -evaluation and minimal number of evaluations was derived.Figures 3-9 display the efficiency  curves showing the common logarithm of the maximum global absolute error throughout the integration versus computational cost measured by time used by each method in the same computation machine.An advantage of the STDRKN methods over the general classical Runge-Kutta-Nyström methods and TDRKN methods is that they can reach higher order with fewer functions evaluations per step and also give us higher stage order than RKN.Some tested problems were performed.From Figures 3,4,5,6,7,8, and 9, the numerical results showed that the new methods agreed very well with  the existing methods in the literature and required less time compared to the existing methods.

Figure 3 :
Figure 3: The time curves for all methods for Problem 1.

Figure 4 :
Figure 4: The time curves for all methods for Problem 2.

Figure 5 :
Figure 5: The time curves for all methods for Problem 3.

Figure 6 :
Figure 6: The time curves for all methods for Problem 4.

Figure 7 :
Figure 7: The time curves for all methods for Problem 5.

Table 3 :
Root trees for STDRKN methods up to order five.