Runge-Kutta-Nystrom method

Second-order differential equations Runge-Kutta-Nystrom method [5] 

This system computes the solution of the initial value problem for an equation y =f x, y, )f), where initial values Xq, y, yo, step size h, and number of steps N are Imown. 

Process engineering and design using Visual Basic 

Solve the following second-order differential equation  

Similarly, other values can be calculated and tabulated as shown in Table 

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In ref. 143 the authors develop a third-order 3-stage diagonally implicit Runge-Kutta-Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems. The obtained method has been developed in order to have minimal local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The authors also study the stability of the method. The new proposed method is illustrated via a set of test problems. 

In ref 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystrom methods with arbitrary high order are developed. The results of this paper are proved based on a S5unmetry argument. 

In ref 164 new and elficient trigonometrically-fitted adapted Runge-Kutta-Nystrom methods for the numerical solution of perturbed oscillators are obtained. These methods combine the benefits of trigonometrically-fitted methods with adapted Runge-Kutta-Nystrom methods. The necessary and sufficient order conditions for these new methods are produced based on the linear-operator theory. 

In ref 171 new Runge-Kutta-Nystrom methods for the numerical solution of periodic initial value problems are obtained. These methods are able to integrate exactly the harmonic oscillator. In this paper the analysis of the production of an embedded 5(3) RKN pair with four stages is presented. 

In ref 174 a new embedded pair of explicit exponentially fitted Runge-Kutta-Nystrom methods is developed. The methods integrate exactly any linear combinations of the functions from the set (exp(/iO,exp(—/it) (ji e ifl or /i e i9I). The new methods have the following characteristics ... 

T. E. Simos and P. S. Williams, A new Runge-Kutta-Nystrom method with phase-lag of order infinity for the numerical solution of the Schrodinger equation, MATCH Commun. Math. Comput. Chem., 2002, 45, 123-137. 

Z. Kalogiratou and T. E. Simos, Construction of trigonometrically and exponentially fitted Runge-Kutta-Nystrom methods for the numerical solution of the SchrSdinger equation and related problems a method of 8th algebraic order, J. Math. Chem., 2002, 31(2), 211-232. 

P. J. Van Der Houwen and B. P. Sommeijer, Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal., 1987, 24(3), 595-617. 

J. M. Franco, Exponentially fitted explicit Runge-Kutta-Nystrom methods, J. Comput. Appl. Math., 2004, 167, 1-19. 

Hans Van de Vyver, On the generation of P-stable exponentially fitted Runge-Kutta-Nystrom methods by exponentially fitted Runge-Kutta methods. Journal of Computational and Applied Mathematics, 2006, 188, 309-318. 

Hans Van de Vyver, A 5(3) pair of explicit Runge-Kutta-Nystrom methods for oscillatory problems, Mathematical and Computer Modelling, 2007, 45, 708-716. 

J. M. Franco, Runge-Kutta-Nystrom methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Commun., 2002, 147, 770-787. 

Hans Van de Vyver, An embedded exponentially fitted Runge-Kutta-Nystrom method for the numerical solution of orbital problems. New Astronomy, 2006, 11, 577-587. 

In this flow chart is the end-point of integration, is the start-point of integration and NSTEP is the number of steps. The computation of qh zu i = 1(1)3 is based on the Runge-Kutta-Nystrom method of Dormand and Prince 8(7) (see 9-10). 

In 52 the author develops a symplectic exponentially fitted modified Runge-Kutta-Nystrom method. The method of development was based on the development of symplectic exponentially fitted modified Runge-Kutta-Ny-strom method by Simos and Vigo-Aguiar. The new method is a two-stage second-order method with FSAL-property (first step as last). 

Phase-lag Analysis of the Runge-Kutta-Nystrom Methods. - For the numerical solution of the problem (129), the m-stage explicit Runge-Kutta-Nystrom (RKN) method shown in Table 6 can be used. Application of this method to the scalar test equation (130) produces the numerical solution... 

Runge-Kutta and Runge-Kutta-Nystrom Methods for Specific Schrodinger Equations... 

Simos, Dimas and Sideridis107 have constructed a Runge-Kutta-Nystrom method with phase-lag of order 8 and with an interval of periodicity equal to (0,9.114). This method is given in Table 12. 

Based on the results presented in the relative papers and based on some numerical tests made for this review, the most efficient Runge-Kutta method for specific Schrodinger equations is the one developed by Simos and Williams106 with seven stages while the Runge-Kutta-Nystrom method developed by Simos, Dimas and Sideridis107 gives similar results in accuracy and computational efficiency. 

Table 12 Runge-Kutta-Nystrom method with phase-lag of order 8...

In [15], [26], [32], [59]-[60], [72] some modified Runge-Kutta or Runge-Kutta-Nystrom methods are constructed. The modification is based on exponential and trigonometric fitting or phase-fitting property. 

In [160] the authors obtained a new fourth algebraic order Rimge Kutta-Nystrom method with vanished phase-lag, amplification error and the first derivatives of the previous properties. More spedlically the authors consider the general form of the Runge-Kutta-Nystrom method... 

In the above seheme if Ci = 0 then we an explicit Runge Kutta Nystrom method. If = 1 and = fc, for j= i then we have an FSAL explicit RKN... 

In [190] the authors studied extended Runge-Kutta-Nystrom methods of the form ... 

In the same formula, h is the step size of the integration and n is the number of steps, i.e. is the approximation of the solution in the point x and Xn = XQ + n h and xq is the initial value point. Using the above methods and based on the theory of phase-lag and amplification error for the Runge-Kutta-Nystrom methods, the authors have derived two fourth algebraic order Runge-Kutta-Nystrom methods with phase-lag of order four and amplification error of order five. For both of methods the authors have obtained the stability regions. The efficiency of the produced methods is proved via numerical experiments. 

The following (m + l)-stage modified Runge-Kutta-Nystrom method has been introduced in order to solve numerically the above problem ... 

As Aguiar, Simos and Tocino have proved, in order that the Runge-Kutta-Nystrom method defined by (5)-(7) integrates exactly equation (3) with /( j = 0 the coefficients A, B, C, D and C Di of the method must be given by ... 

Runge-Kutta-Nystrom Method with FSAL Property. - We give the following definition. 

For w 0, the above method is equivalent to the classical symlpectic second algebraic order Runge-Kutta-Nystrom method mentioned in the paper of Calvo and Sanz-Serna. ... 

Modified Runge-Kutta-Nystrom Phase-fitted Methods. - Simos et al. have considered a modified four-stage explicit Rxmge-Kutta-Nystrom method presented in Table 14. The free parameter 3 is determined in order that the method has phase-lag of order infinity (see ref. 6 for more details on phase-lag analysis of Runge-Kutta-Nystrom methods). 

Table 14 The modified four-stage explicit phase-fitted Runge-Kutta-Nystrom method of Simos et al.

In [184] the authors are studied the Runge-Kutta-Nystrom method of the form... 

J. R. Cash and S. Girdlestone, Variable Step Runge-Kutta-Nystrom Methods for the Numerical Solution of Reversible Systems, JNAIAM J. Numer. Anal. Indust. Appl. Math, 2006, 1(1), 59-80. 

T. E. Simos, E. Dimas and A. B. Sideridis, A Runge-Kutta-Nystrom Method for the Numerical-Integration of Special 2nd-order Periodic Initial-Value Problems, Journal of Computational and Applied Mathematics, 1994, 51(3), 317-326. 

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