Abstract
Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions and respectively, when , the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.
1. Introduction
In this study, we consider the numerical solution of the general second-order initial value problems (IVPs) of the following form:where is a continues vector valued functions, and is the dimension of the system. Problems of form (1) occur in different areas of applied science such as quantum mechanics, classical mechanics, celestial mechanics, circuit theory, astrophysics, control theory, and biological sciences (see Dormand [1], Hairer et al. [2], and Kristensson [3]) which attract the interest of several researchers (see [4–11]). Many well-known researchers have constructed Runge-Kutta-Nyström methods (RKN) for solving (1) by using numerous techniques, for instance, exponentially fitted and trigonometrically fitted techniques. Simos [12] the first author studied exponentially fitted RKN method in the literature. Franco [13] derived exponentially fitted RKN methods with three and four orders. Van de Vyver [14] constructed the embedded pair of exponentially fitted RKN methods to solve orbital problems. Yang et al. [15] derived trigonometrically fitted adapted Runge-Kutta-Nyström methods (ARKN) with perturbed oscillators. Fang et al. [16] presented embedded pair trigonometrically fitted ARKN methods. Senu et al. [17] derived zero-dissipative RKN method with minimal phase-lag for solving special second-order ODEs with oscillatory solutions. Moo et al. [18] constructed two new phase-fitted and amplification-fitted RKN based on Garcia and Hairer methods of orders four and five, respectively. Recently, Fang et al. [19] proposed a block hybrid trigonometrically fitted RKN methods for solving problem (1). Demba et al. [20] derived two trigonometrically fitted RKN methods based on Garcia and Hairer methods of orders four and five, respectively, for periodic IVPs. Demba et al. [21] constructed a pair of embedded trigonometrically fitted RKN method for solving second-order IVPs where the solutions are oscillatory. Ehigie et al. [22] derived two implicit TDRKN methods via mixed collocation.
There are no research findings related to exponentially fitted and trigonometrically fitted TDRKN methods in which researchers have not yet explored the advantages or disadvantages in applying exponentially fitting and trigonometrically fitting techniques to TDRKN methods. Hence, in this paper, new two exponentially fitted and trigonometrically fitted TDRKN methods are constructed. In Section 2, explicit exponentially fitted two-derivative Runge-Kutta-Nyström methods with two-stage fourth order and three-stage fifth order are derived. In Section 3, trigonometrically fitted explicit TDRKN methods with two-stage fourth order and three-stage fifth order were derived. In Section 4, the convergence and linear stability analysis is analyzed. Numerical illustrations are presented in Section 5 to show the efficiency of the new methods. Finally, the discussion and conclusion are given in Section 6.
2. Exponentially Fitted TDRKN Methods
In this section, we will determine the conditions of exponentially fitted two-derivative Runge-Kutta-Nyström methods. We suppose that the third derivative for (1) is known. An -stage TDRKN method for (1) is defined by the following formula (see Chen et al. [10]):where are real numbers. To construct exponentially fitted TDRKN method it is needed at each stage to integrate exactly the function and ; therefore the following four equations are obtained:and four more equations corresponding to and where , a real number and indicates the frequency of the problem and is the step-length of integration. The relations , will be used in the derivation process. The following conditions are obtained: and four equations corresponding to and :
2.1. Exponentially Fitted TDRKN Method of Order Four
Referring to the following fourth-order two-stage method derived by Chen et al. [10], , , , , , , solving (8) to (11) yields Next, solve (12) to (15) and use the above coefficients to find , , , and : These lead to our new method, an explicit exponentially fitted TDRKN two-stage fourth-order method denoted as EFTDRKN4. The corresponding Taylor series expansion of the solution is given by As , the coefficients of the new method EFTDRKN4 and reduce to the coefficients of the original method TDRKN4. That is to say , and are identical to , and of TDRKN4 method.
2.2. Exponentially Fitted TDRKN Method of Order Five
Referring to the following fifth-order three-stage method derived by Chen et al. [10], , , , , , , , , , , . Solving (8) to (11) yieldsNext, solve (12) to (15) and use the above coefficients to find , , and : These lead to our new explicit exponentially fitted explicit TDRKN three-stage fifth-order method denoted as EFTDRKN5. The corresponding Taylor series expansion of the solution is given by As , the coefficients of the new method EFTDRKN5 and , reduce to the coefficients of the original method TDRKN5.
3. Trigonometrically Fitted TDRKN Methods
Exponentially fitted methods lead to trigonometrically fitted methods: when replacing with , we obtainand four equations corresponding to and :
3.1. Trigonometrically Fitted TDRKN Method of Order Four
Consider the same coefficients fourth-order two-stage method developed by Chen et al. [10], as in Section 2.1. By solving (22) to (25) using the Chen coefficients and letting , , , , and , we obtain the following:Next, solve (26) to (29), and usethe above Chen coefficients to find , , and : These lead to our new trigonometrically fitted explicit TDRKN which is called TFTDRKN4 method. The corresponding Taylor series expansion of the solution is given by
3.2. Trigonometrically Fitted TDRKN Method of Order Five
Consider the same coefficients fifth-order three-stage method developed by Chen as in Section 2.2. By solving (22) to (25) and letting , , , , , , and as free parameters, we obtain Next, solving (26) to (29) and using the above Chen coefficients to find , ,, and yield These lead to our new explicit trigonometrically fitted which is called TFTDRKN5 method. The corresponding Taylor series expansion of the solution is given by
4. Analysis of Convergence and Stability of the Method
In this section, the convergence and linear stability of the new methods derived will be discussed. Note that zero-stability and consistency are sufficient conditions for the new method to be convergent. The method (2) is said to be consistent if it has at least order two and every one-step method is always zero-stable (see Lambart [[23]]). Hence, the proposed method is zero stable and consistent; as a result, the proposed method can be said to be convergent. For the linear stability, consider the test equation: is the frequency of the problem. Applying method (2) to (36) yields where and . Matrix is called stability matrix and and are defined as given in [10]. The absolute stability regions of the new methods are presented as follows (see Figures 1–4 for EFTDRKN4, TFDDRKN5, TFTDRKN4 and TFTDRKN5 methods, respectively).
5. Problems Tested and Numerical Results
In order to examine the effectiveness of the new TFTDRKN and EFTDRKN methods proposed in this paper, we apply them to some test problems. We also employ several integrators from the literature for comparison. The numerical methods we choose for experiments are as follows:(i) EFTDRKN5(3): the three-stage fifth-order exponentially fitted TDRKN method derived in this paper.(ii) EFTDRKN4(2): the two-stage fourth-order exponentially fitted TDRKN method derived in this paper.(iii) TFTDRKN5(3): the three-stage fifth-order trigonometrically fitted TDRKN method derived in this paper.(iv) TFTDRKN4(2): the two-stage fourth-order trigonometrically fitted TDRKN method derived in this paper.(v) TDRKN5(3): the three-stage fifth-order trigonometrically fitted TDRKN method presented in [10].(vi) TDRKN4(2): the two-stage fourth-order trigonometrically fitted TDRKN method presented in [10].(vii) EFRKS4: the exponentially fitted classical RK method derived in [12].(viii) EFRKF4(5): the five stage fourth-order trigonometrically fitted RK method derived in [24].(ix) RKF5(6): the six-stage fifth-order Fehlberg RK method derived in [1].(x) DOPRI5: the six-stage fifth-order RK method derived in [1].(xi) PFAFRKC4(5): the five-stage fourth-order phase-fitted and amplification-fitted RK method given in [25].(xii) PFAFRKF4(5): the five-stage fourth-order phase-fitted and amplification-fitted modified RK method given in [26].(xiii) RKNG5(6): the classical six-stage fifth-order RKN method which is the limit method of ARKNGV5 as the frequency matrix derived in [7].(xiv) PFAFRKS5(6): the six-stage fifth-order phase-fitted modified RK method given in [27].(xv) PFAFRKC5(6): the six-stage fifth-order phase-fitted and amplification-fitted RK method given in [25].(xvi) RK6(8): the classical eight-stage sixth-order RK method derived in [28].
Problem 1. We consider the linear equation studied in [29]: with the initial values The problem is integrated on the interval with stepsizes and . The analytical solution of this problem is given by the following:
Problem 2. We consider the inhomogeneous linear equation: with the initial values The problem is integrated on the interval with stepsizes and . The analytical solution of this problem is given by:
Problem 3. We consider the inhomogeneous linear system: with the initial values The analytical solution of this problem is given by the following: The problem is integrated on the interval with stepsizes and
Problem 4. We consider the inhomogeneous linear system: with the initial values The analytical solution of this problem is given by the following: The problem is integrated on the interval with stepsizes and
Problem 5. We consider the harmonic oscillator equation with frequency and small perturbation that is studied in [30]: with the initial values The analytical solution of this problem is given by the following: In this experiment we choose the parameters values , . The problem is integrated on the interval and the stepsizes are taken as and
Problem 6. We consider the famous Van der Pol equation studied in [31]: with the initial valuesThis is a nonlinear equation. Here we take . The problem is integrated on the interval with stepsizes and Since the exact solution to the problem is not available, when estimating the error of each method, we use RKNG4 method in [2] as a reference numerical solution with a very small stepsize.
Problem 7. We consider the inhomogeneous linear system: with the initial values The analytical solution of this problem is given by the following: The problem is integrated on the interval and the stepsizes are taken as and
Problem 8. We consider the inhomogeneous linear system: with the initial values The analytical solution of this problem is given by the following: The problem is integrated on the interval and the stepsizes are taken as and
Problem 9. Consider the linear nonhomogeneous system (see [31]): with the initial valuesThe problem is integrated on the interval with stepsizes .
The exact solution is as follows:
Problem 10. Consider the damped wave equation with periodic boundary conditions (see [31]): A semidiscretization in the spatial variable by second-order symmetric differences leads to the following system of second-order ODEs in time: where with .with and and . In this experiment, we take and and the initial conditions as We choose and integrate the problem in the interval with the stepsizes . The reference numerical solution is obtained by method RKNG4 with a very small stepsize.
6. Discussion and Conclusion
An analysis of the construction of a class of exponentially fitted and trigonometrically fitted two-derivative Runge-Kutta-Nyström methods for solving has been carried out.
The efficiency of the methods developed is presented in Figures 5–28 by plotting the graph of the decimal logarithm of the maximum global error against the logarithm number of function evaluations. Observing from the graph plotted in Figures 5–12, we noticed that, for all problems, the proposed methods EFTDRKN4(2) and EFTDRKN5(3) are more efficient compared to TDRKN methods of the same order and other existing methods in the literature.
From Figures 13, 15, 17, 18, 19, 21, 23, 25, and 27, we observed that, for Problems 5–9, the TFTDRKN4(2) and TFTDRKN5(3) methods have better accuracy than the other methods and good for oscillatory problems. In Figures 26 and 28, the new methods TFTDRKN4(2) and TFTDRKN5(3) have the same order of accuracy with the classical TDRKN methods. However, the results for Problems 6 and 10 given in Figures 16, 22, 26, and 28 showed that TFTDRKN4(2) and TFTDRKN5(3) methods can be used to solve accurately problems without exact solutions.
Figures 14, 16, 20, and 22 show the effect of trigonometrically fitting approach over the classical TDRKN methods when certain selected value is considered and from figures it is clearly shown that the global error increases slowly compared to original TDRKN which rapidly increases along the integration process.
In this paper, two exponentially fitted explicit two-derivative RKN methods which are suitable for problems with exponential solutions and also two trigonometrically fitted explicit two-derivative RKN methods suitable for oscillating solutions especially for long-range integration have been derived. From the numerical results, it can be concluded that the new methods are more promising compared to standard TDRKN and with other existing methods in the literature.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors have declared that no conflicts of interest exist
Acknowledgments
The authors gratefully acknowledge the financial support of Universiti Putra Malaysia.