New Phase-Fitted and Amplification-Fitted Fourth-Order and Fifth-Order Runge-Kutta-Nyström Methods for Oscillatory Problems

and Applied Analysis 3 and zero amplification error which is based on Garcia’s RKN method of three-stage and algebraic order four as follows


Introduction
Throughout this paper, we are dealing with the initial value problems (IVP) related to second-order ODEs in the form: where  is independent of   () explicitly and it is known that their solutions are periodic.This kind of problems often arise in many fields of applied sciences such as astronomy, quantum mechanics, physical chemistry, structural mechanics, and electronics.Over the last three decades, there are a number of numerical methods that have been derived by several authors based on different approaches such as minimal phase-lag, phase-fitted, and exponential-fitted for solving second-order oscillatory IVPs.See van der Houwen and Sommeijer [1], Senu et al. [2], van de Vyver [3], Vigo-Aguiar and Ramos [4], Ramos and Vigo-Aguiar [5], and Ismail [6].The term phaselag was first introduced by Brusa and Nigro [7] in 1980.In 1993, Simos derived a Runge-Kutta-Fehlberg method based on the idea of phase-lag of order infinity [8].Recently, the idea of phase-lag of order infinity has been used to developed new numerical methods.Simos and Vigo-Aguiar presented in [9] a modified Runge-Kutta-Nyström method with phaselag of order infinity for solving the Schrödinger equation and related problems.Papadopoulos et al. [10] presented a new phase-fitted RKN method for solving IVP with oscillating solutions.
In this paper, we will combine the idea of phase-lag of order infinity and zero amplification error together.First, we will construct a three-stage phase-fitted and amplificationfitted RKN method which is based on Garcia's method of algebraic order four [11].Then, we will construct a four-stage phase-fitted and amplification-fitted RKN method based on the coefficients of Hairer's method of algebraic order five [12].

Analysis Phase-Lag of the Methods
The general -stage RKN method for (1) is of the form where In the investigation of phase-lag error of the method, we will use the test equation in the following: By applying method (2) to test equation ( 4), it yields where , ,   , and   are polynomial in term of  2 and are totally determined by the parameters of method (2).It is given that the exact solution of ( 4) is where By substituting (7) into (6), it yields Then, we assume that the eigenvalues of  are  1 ,  2 , and they will be called as the amplification factors of the RKN method.The consequent eigenvectors are [1,  1 ]  , [1,  2 ]  , where   =   /(  −   ),  = 1, 2. The numerical solution of ( 4) is where If  1 and  2 are complex conjugate, then  1,2 = || exp(±) and  1,2 = || exp(±).By substituting both into (9), we have From the exact solution (8) and the numerical solution (11), we can forward to the following definition.
Let us denote that From Definition 1 it follows that ) ,          = √  ( 2 ).( 13) From ( 13), we have the following definition for the case of phase-lag of order infinity.
Definition 3 (phase-lag of order infinity, see [10]).To obtain phase-lag of order infinity the relation When at a point , where () = 0, the method is said to have zero amplification error (zero-dissipative), and we have From the condition above, we have the following remark.
Remark 4 (see [13]).To achieve phase-lag of order infinity and zero amplification error, the relations below must hold

Construction of the New Methods
In this section, we will present the construction of two new RKN methods.The first method is based on a three-stage RKN method with algebraic order four (see [11]).The second method is based on a four-stage RKN method with algebraic order five (see [12]).

New Fourth Order RKN Method.
In this section, we want to derive an RKN method with phase-lag of order infinity and zero amplification error which is based on Garcia's RKN method of three-stage and algebraic order four as follows First of all, we have to compute the polynomials , ,   , and   and substitute all of them into (12).Hence, we have  and  in terms of parameters of RKN method.In order to construct the new method with phase-lag of order infinity and zero amplification error, we set  31 and  32 as free parameters but leave the rest of the coefficients.Then we substitute the value of the coefficients above into  and  yielding  ( As mentioned in Remark 4, in order to achieve phase-lag of order infinity and zero amplification error, conditions (15) must be satisfied.Therefore, we substitute  and Q from ( 17) into ( 15) and solve the equations yielding For small value of , we use the Taylor series expansion as follows: (21) Then we substitute (21) into (15) and solve the equations yielding For small value of , we use the Taylor series expansion as follows: Thus, we have obtained a new method which has two variable coefficients,  21 and  31 , that depend on the product of the step-size ℎ and the frequency of the method V. We denote this method as PFAFRKN5.For linear problems, the frequency of the method is the same as the frequency of the exact solution.For nonlinear problems, however, the frequency of the method is usually different from the frequency of the exact solution but depends on the method itself (see [14]).For each specific product of Vℎ, it helps to nullify the phase-lag error and the amplification error.For solving oscillatory problems, reducing the phase-lag error of the method is far more important than decreasing its algebraic error.

Algebraic Order and Error Analysis
In this section, we will compute the local truncation error (LTE) of the new methods and verify the algebraic order of the methods.
From (26), method PFAFRKN5 is said to have algebraic order 5 since all of the coefficients up to ℎ 5 vanished.Table 1 shows a comparison of the properties of the methods derived.

Problems Tested and Numerical Results
In this section, we will apply the new methods to some second-order differential equation problems.The following explicit RKN methods are selected for the numerical comparisons.Fourth-order: (i) PFAFRKN4: the new derived fourth-order RKN method; (ii) RKN4G: the three-stage fourth-order RKN method derived by García et al. [11]; (iii) SRKN3V: the third-order symplectic RKN method with minimal phase-lag derived by van de Vyver [3]; (iv) PFERKN4P: the fourth-order RKN method with phase-lag of order infinity derived by Papadopoulos et al. [10]; (v) OPTRKN4P: the fourth-order optimized RKN method derived by Papadopoulos and Simos [13]. Fifth-order: (i) PFAFRKN5: the new derived fifth-order RKN method; (ii) RKN5H: the four-stage fifth-order RKN method derived by Hairer et al. [12]; (iii) OPTRKN5K1: the fifth-order optimized RKN method with zero first derivative of phase-lag derived by Kosti et al. [15]; (iv) OPTRKN5K2: the fifth-order optimized RKN method with zero first derivative of amplification factor derived by Kosti et al. [16].
Those methods are categorized into two categories according to algebraic order of each method for comparison purposes.The accuracy criteria taken is calculating the log 10 of the maximum absolute error, where   =  0 + ℎ,  = 1, 2, . . ., ( −  0 )/ℎ.We test the problems in the following for the interval of [0, 4000] with the step-size ℎ = 1/2  ,  ≥ 1.
Estimated frequency: V = 20.Source: Lambert and Watson [18].The numerical results are plotted in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16.Figures 1-4 and Figures 9-12 display the efficiency curves where the common logarithm of the maximum global absolute error throughout the integration versus computational cost measured by number of function evaluations.Figures 5-8 and Figures 13-16 display the efficiency curves where 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.Figures 1-8 show the efficiency curves of the methods in the first category that consist of PFAFRKN4, RKN4G, SRKN3V, PFERKN4P, and OPTRKN4P, whereas Figures 9-16 show the efficiency curves of the fifth-order methods consist of PFAFRKN5, RKN5H, OPTRKN5K1, and OPTRKN5K2.These efficiency curves display a clear comparison among those methods.
From Figures 1-8, we found that the new method PFAFRKN4 is the most accurate and efficient for solving Problems 1-5, followed by OPTRKN4P, SRKN3V,   PFERKN4P, and the corresponding original method RKN4G, whereas for the fifth-order methods, from Figures 9-16, we observed that the new method PFAFRKN5 is the most efficient method, followed by optimized RKN methods OPTRKN5K1 and OPTRKN5K2 and the corresponding original method RKN5H.In addition, we can observe that the new methods are having the same computational cost with the corresponding original methods.

Conclusion
In this paper, we have derived two new phase-fitted and amplification-fitted RKN methods for solving second-order IVPs which are oscillatory in nature.First method is based on Garcia's fourth algebraic order RKN method and the second method is based on Hairer's four-stage fifth algebraic order RKN method.Numerical results show that both methods are more accurate and efficient for solving second-order differential equations with oscillating solutions.
In this case, we set  21 and  31 as free parameters and we found that  and  are Hence, a new method is derived and we denote it by PFAFRKN4.This method has two variable coefficients  31 and  32 that depend on the product of the step-length ℎ and the frequency of the method V.3.2.New Fifth Order RKN Method.In this section, we want to derive a new RKN method which is based on Hairer's RKN method of four-stage and algebraic order five as follows:

Table 1 :
Summary of the properties of the methods.