Phase Fitted Methods

3 Phase Fitted Methods. - Simos45 has introduced the following one-parameter sixth algebraic order method ( )  

Raptis and Simos33 have constructed the following one-parameter family of four-step sixth algebraic order methods  

In the same paper a direct formula for the calculation of the phase-lag for symmetric four-step methods is obtained. Based on this formula the free parameter cq of the above method is obtained in order the phase-lag order is equal to infinity. 

Simos46 has derived the following two symmetric two-step methods of algebraic order four and six. Both methods contain one free parameter. 

The free parameter c of the above methods is determined in order the phase-lag to be of order infinity. We note that the methods are almost P-stable, i.e. they have interval of periodicity equal to (0, oo) — D where D is a set of distinct points. 

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T. E. Simos, Predictor-corrector phase-fitted methods for y = i(x,y) and an applica-... 

A. D. Raptis and T. E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problems, BIT, 1991, 31, 160-168. 

A New Phase Fitted Method. - Consider the one free parameter symmetric three-step hybrid predictor-corrector explicit method ... 

In Table 4 we present the properties of the phase fitted methods. 

Numerical Illustrations for Exponentially-Fitted Methods and Phase Fitted Methods. - In this section we test several finite difference methods with coefficients dependent on the frequency of the problem to the numerical solution of resonance and eigenvalue problems of the Schrodinger equations in order to examine their efficiency. First, we examine the accuracy of exponentially-fitted methods, phase fitted methods and Bessel and Neumann fitted methods. We note here that Bessel and Neumann fitted methods will also be examined as a part of the variable-step procedure. We also note that Bessel and Neumann fitted methods have a large penalty in a constant step procedure (it is known that the coefficients of the Bessel and Neumann fitted methods are position dependent, i.e. they are required to be recalculated at every step). 

Table 4 Properties of phase-fitted methods. S = H2 H = sqn, q = 1,2,.... A.O. is the algebraic order of the method. Int. Per. is the interval of periodicity of the method. N.o.S. is the number of steps of the method. I = Implicit, E - Explicit.rev This review...

Remarks and Conclusion. - (I) Resonance Problem For the resonance problem the most accurate methods are the methods derived by Simos and Williams (Case IV, Case V and Case VI of the family),20 the method derived by Simos,24 the hybrid sixth algebraic order methods derived by Thomas and Simos (Case IV and Case V of the family),25 the new phase fitted method of algebraic order eight developed in this critical review in Section 2.3.1, the eighth algebraic order exponentially-fitted method derived by Simos,30 the eighth... 

Avdelas and Simos,93 (13) the exponentially-fitted variable-step method developed by Simos,8 (14) the variable-step phase-fitted method developed by Simos,51 (15) the variable-step P-stable method developed by Simos,74 (16) the exponentially-fitted variable-step method developed by Thomas and Simos,25 (17) the variable-step Bessel and Neumann fitted method developed by Simos,43 (18) the variable-step Bessel and Neumann fitted method developed by Simos,44 (19) the new exponentially-fitted variable step method based on the new exponentially-fitted tenth algebraic order method developed in Section... 

Modified Runge-Kutta Phase-fitted Methods. - Simos et al. have considered a modified four-stage exphcit Runge-Kutta method presented in Table 12. The parameters of the method are given by ... 

Third methodology for the development of numerical methods Phase-fitted methods... 

Based on the Figs. 4, 5 and 6 we can say the both of trigonometrically fitted and Phase-fitted methods are not P-stable (i.e. there are areas in the Figs 4, 5 and 6 that are white and in which the conditions of P-stability are not satisfied)... 

Fig. 6 w-H plane of the Phase-Fitted method of the new family of methods produced in section (2.3) (a) Method presented in paragraph 2.1 (b) Method presented in paragraph 2.2 (c) Method presented in paragraph 2.3. 

In the case where the frequency of the exponential fitting is equal to the frequency of the scalar test equation, the intervals of periodicity are equal to (0, n ) for the Trigonometrically-Fitted Method produced by first methodology in section 2.1, (0, 3.486913415)-S for the Trigonometrically-Fitted Method produced by second methodology in section 2.2 (where S is equal to 0.01252513416) and finally (0, 3.986068639)-Q for the Phase-Fitted Method produced by second methodology in section 2.3 (where Q is equal to 0.7854483473)... 

Phase-Fitted method, whieh is indicated as Method D, have the same behavior. 

T. E. Simos, An Explicit 4-Step Phase-Fitted Method for the Numerical-Integration of 2nd-order Initial-Value Problems, Journal of Computational and Applied Mathematics, 1994, 55(2), 125-133. 

T. E. Simos, Predictor Corrector Phase-Fitted Methods for Y" = F(X,Y) and an Application to the Schrodinger-Equation, International Journal of Quantum Chemistry, 1995, 53(5), 473 3. 

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