Runge-Kutta Exponentially Fitted Methods

3 Runge-Kutta Exponentially Fitted Methods. - Williams and Simos have considered the four-stage explicit Runge-Kutta method presented in Table 9, where the parameters of the method are given by  

Wilhams and Simos have determined the free parameter in order that the 

Simos has constructed a four-stage fourth algebraie order explicit Runge-Kutta exponentially fitted and trigonometrically fitted method of the form presented in Table 9. The parameters of the method are determined in order that 

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In Section 5 trigonometrically fitted Fifth algebraic order Runge-Kutta methods are presented. For these methods we present the construction and the error analysis from which one can see that the classical method is dependent on the third power of energy, the first exponentially-fitted methods is depended on a second power of the energy and finally the second exponentially-fitted methods is depended on first power of the energy. 

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). 

The construction of Runge-Kutta and Runge-Kutta-Nystrom exponentially-fitted methods. 

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 161 the authors have produced exponentially-fitted BDF-Runge-Kutta type formulas (of second-, third- and fourth-order). The good behaviour of the new produced methods for stilf problems is proved. The stability regions of the new proposed methods have been examined. It is proved that the plots of their absolute stability regions include the whole of the negative real axis. Finally the author propose several procedures to find the parameter of the new methods. 

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

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, Exponentially-Fitted Runge-Kutta Third Algebraic Order Methods for the Numerical Solution of the SchrSdinger Equation and Related Problems, International Journal of Modern Physics C, 1999, 10(5), 839-851. 

Z. A. Anastassi and T. E. Simos, A family of exponentially-fitted Runge-Kutta methods with exponential order up to three for the numerical solution of the Schrodinger equation, J. Math. Chem., 2007, 41(1), 79-100. 

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

Exponentially Fitted Runge-Kutta Methods. - The method (41) is associated with the operator,... 

In chapter 3, the construction of exponential fitting formulae is presented. In chapter 4, applications of exponential fitting to differentiation, to integration and to interpolation are presented. In chapter 5, application of exponential fitting to multistep methods for the solution of differential equations is presented. Finally, in chapter 6, application of exponential fitting to Runge-Kutta methods for the solution of differential equations is presented. 

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 [209] the authors studied the optimization of embedded explicit exponentially-fitted Runge-Kutta methods. The explicit exponentially-fitted Runge-Kutta methods described in the paper was written as ... 

Table 9 The four-stage explicit exponentially fitted and trigonometrically fitted Runge-Kutta method of Williams and Simos...

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

Z. Kalogiratou, Th. Monovasilis and T. E. Simos, Computation of the Eigenvalues of the Schrodinger Equation by Exponentially-Fitted Runge-Kutta-Nystrom Methods, Computer Physics Communications, 2009, 180(2), 167-176. 

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