P-Stable Methods

1 Implicit Methods. - 4.1.1 P-stable Methods. - 4.1.1.1 Fourth algebraic order methods. - Simos and Raptis64 have constructed P-stable methods with minimal phase-lag for the numerical solution of Schrodinger type equations. It is the first article in the literature in which a combination of the properties of P-stability and phase-lag has been obtained in order to construct methods for Schrodinger type equations. They have produced the following two families of methods  

The parameters a, b(, /=1,2 are computed in order to satisfy the two properties, i.e. the minimal phase-lag property and the P-stability property. As a result of the above, two P-stable methods are obtained. One for / = 1 which is a fourth algebraic order P-stable method with phase-lag of order eight and another for i = 2 which is a fourth algebraic order P-stable method with phase-lag of order ten. 

Simos and Mousadis65 have considered the following family of fourth algebraic order methods 

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In the present chapter we have presented a new family of exponentially-fitted four-step methods for the numerical solution of the one-dimensional Schrodinger equation. For these methods we have examined the stability properties. The new methods satisfy the property of P-stability only in the case that the frequency of the exponential fitting is the same as the frequency of the scalar test equation (i.e. they are singularly P-stable methods). The new methods integrate also exactly every linear combination of the functions... 

The Method of Raptis and Allison" is more efficient than the Numerov s method and the P-stable method of Chawla ... 

In ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

G. Avdelas and T. E. Simos, A generator of high-order embedded P-stable method for the numerical solution of the Schrodinger equation, J. Comput. Appl. Math., 1996, 72, 345-358. 

Qinghong Li and Xinyuan Wu, A two-step explicit P-stable method of high phase-lag order for linear periodic IVPs, Journal of Computational and Applied Mathematics, 2007, 200, 287-296. 

In Section 4 we present Four-step P-stable Methods with minimal Phase-Lag. We give a new procedure for the construction of such methods. This procedure is based on the requirement that the roots of the characteristic equation associated with the methods must have specific forms. For these methods numerical results on the resonance problem of the radial Schrodinger equation are given and analysed. 

Four-Step P-Stable Methods with Minimal Phase-Lag... 

New developed four-step P-stable method with minimal phase-lag (indicated as Method [g]). 

In 60 the author presents a modified Numerov P-stable method. The construction of this method is based on the procedure mentioned above (see comment on 53). 

The free parameter cq is determined in order the phase-lag order to be equal to infinity. We note the method is an explicit almost P-stable method. 

The values of parameters a and b are obtained in order to construct P-stable methods with minimal phase-lag. First the parameter b is determined in order to have phase-lag of order eight and then the condition of parameter a in order that the P-stability property is satisfied is obtained. Based on the above procedure Simos has proved that all the methods of the above family with a > t j are P-stable methods with phase-lag of order eight. In the same paper Simos has proved that if parameter a is equal to — then a method... 

The parameters , / = 0(1)2 are obtained in order that P-stable methods with minimal phase-lag be constructed. First the parameters a and ai are determined in order to have phase-lag of order ten. Then the condition for the... 

Table 7 Properties of P-stable methods. A.O. is the algebraic order of the method. P.L.O. is the phase-lag order, Int. Per. is the interval of periodicity of the method. N.o.S. is the number of steps of the method. All the methods are Implicit. We note that the modifications of methods included in this Table are developed in this review.

Simos74 has considered two modifications of the P-stable method of Simos.67 In the first modification an extra layer in the top of the algorithm is used. The values of parameters a, b, c are obtained in order to produce a P-stable method with minimal phase-lag. As a result of the above, all the methods of the family... 

Based on the theory fully described in refs. 95 and 96 two P-stable methods of exponential and phase-lag order eight and ten are produced. Based on these methods a new variable-step algorithm is introduced. 

Based on the above family of methods Avdelas and Simos97 have given formulae for the coefficients of the P-stable method of exponential order 2m + 2 for arbitrary m. We note that the coefficients are determined in the computer automatically. The new approach is called generator of methods. This was the first generator of P-stable methods in the literature. We also note that the term generator is first introduced in this paper. 

Simos98 has derived another family of P-stable methods of exponential order 2m -l- 2 for m = 3(1)6. The methods here are of hybrid type and have the form ... 

P-stable method of exponential order 10 is produced. For the values... 

Remarks and Conclusion. - The most accurate fourth order P-stable method with constant coefficients is the P-stable method proposed by Simos66 with phase-lag of order sixteen. The most accurate sixth order method with constant coefficients is the modification of the family of sixth order methods of Simos69 with phase-lag of order twenty-two developed in this review. The most accurate eighth order method with constant coefficients is the eighth order method of Simos77 with phase-lag of order eighteen and interval of periodicity... 

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... 

The P-stable method is equivalent to the method with trigonometric order one... 

S. Stavroyiannis and T. E. Simos, Optimization as a Function of the Phase-Lag Order of Nonlinear Explicit Two-Step P-Stable Method for Linear Periodic IVPs, Applied Numerical Mathematics, 2009, 59(10), 2467-2474. 

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