Sixth Algebraic Order Methods

2 Sixth Algebraic Order Methods. - Simos83 has considered the following three free parameters family of explicit sixth algebraic order methods  

Simos84 has derived the explicit version of the sixth algebraic order method developed in ref. 67. Here the method does not have the first layer. The value of parameter a is determined in order to satisfy the minimal phase-lag property. As a result of this, the family of methods with a = has phase-lag of order eight and interval of periodicity equal to (0,21.48). 

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In Table 2 we present the basic properties of the sixth algebraic order methods. 

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 ... 

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... 

Sixth Algebraic Order Methods. - Simos67 has constructed the following family of hybrid sixth algebraic order predictor-corrector methods ... 

Simos92 has considered families of sixth algebraic order methods. These families are based on the formula... 

G. Psihoyios and T. E. Simos, Sixth algebraic order trigonometrically fitted predictor-corrector methods for the numerical solution of the radial Schrodinger equation, J. Math. Chem., 2005, 37(3), 295-316. 

In 26 the authors have developed a new trigonometrically-fitted predictor-corrector (P-C) scheme based on the Adams-Bashforth-Moulton P-C methods. In particular, the predictor is based on the fifth algebraic order Adams-Bashforth scheme and the corrector on the sixth algebraic order Adams-Moulton scheme. More specifically the new developed scheme integrates exactly any linear combination of the functions ... 

Table 2 Properties of two-step sixth algebraic order exponentially-fitted methods. S = H2 H = sqn, q — 1,2,.... The quantities m and p are defined by (11). 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.E.F = Integrated Exponential Functions. SiMi = Simos and Mitsou.22 TS = Thomas and Simos25...

Simos27,28 has derived explicit four-step sixth algebraic order almost P-stable exponentially-fitted methods of the form ... 

Simos43 has derived the following two-parameter sixth algebraic order symmetric two-step predictor-corrector method ... 

More specifically, we will consider a four-step method of sixth algebraic order developed by Wang [68]. Based on method we will develop an optimized method which will have phase-lag and its first derivative equal to zero. We will investigate the stability and the error of the produced methods. Finally, we will apply both of methods to the resonance problem of the radial Schrodinger equation. This is one of the most difficult problems arising from the one-dimensional Schrodinger equation. The paper is organized as follows ... 

This request leads to the determination of the coefficients of the method (46), i.e. with the above request has detrmined the coefficients uo, a, bo, b while the following also relation is hold oo = aj bo = bj. Using the new obtained coeffients the author has proved that the method (46) with coefficients ai,bi, i = 0(1)2 obtained in the paper [161] is a trigonometrically-fitted method which is accurate for the set of functions (47) and is of sixth algebraic order. The author applied the new obtained method to the Hamiltonian system ... 

We consider the multistep symmetric method of Jenkins, with six steps and sixth algebraic order ... 

For all the above cases the authors produced a local truncation error analysis. From this analysis it has been concluded that the exponentially-fitted one-step Obrechkoff method of the third exponential order is the most accurate one. In the same paper a stability analysis was given. From this analysis it has been summarized that the new exponentially-fitted one-step Obrechkoff methods are P-stable. A phase-lag analysis was also presented. The conclusion was that the new developed methods are of sixth phase-lag order (i.e. the same as the algebraic order). Numerical results have shown that the new one-step Obrechkoff methods developed in this paper was much more accurate than multistep methods. This is very interesting since it was known, generally, that one-step methods was less efficient than the corresponding multistep formulae. 

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