Eighth Algebraic Order Methods

3 Eighth Algebraic Order Methods. - Simos85 has derived the explicit version of the eighth algebraic order method derived by Simos in ref. 77. Here the parameters are a,-, i — 1(1)3. The free parameters at, i = 1(1)3 are defined in order to minimize the phase-lag. As a result of the above, the method where a = 0, 2 = — 5 and a = — 2 as phase-lag of order twelve and an interval of periodicity equal to (0,12.9394). Also, the method where 

As a result of the above, the method where b = 0.003149214873243 and a = — ) ] 0 — has phase-lag of order twelve and interval of periodicity 

Recently, Simos and Tsitouras90 have developed an eighth order explicit method which has the form  

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T. E. Simos and J. Vigo-Aguiar, Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrodinger equation, J. Math. Chem., 2002, 31(2), 135-144. 

Eighth Algebraic Order Methods. - Consider the following family of eighth algebraic order methods ... 

Table 2 Coefficients of the eighth algebraic order method of Tsitouras with phase-lag of order 22 and dissipation order 10...

Avdelas et have introduced a generator of hybrid explicit eighth algebraic order method of the form ... 

Again the free parameters of the method wp = O, 11 are defined in order for the method to be exaet for the spherical Bessel and Neumaim functions respectively. Simos has constructed the following implicit eighth algebraic order method ... 

J. Vigo-Aguiar and T. E. Simos, A family of P-stable eighth algebraic order methods with exponential fitting facilities. Journal of Mathematical Chemistry, 2001, 29(3), 177-189. 

The methods are called multiderivative since uses derivatives of order two, four or six. The parameters of the method are computed in order to have eighth algebraic order and minimal phase-lag. Finally, a family of eighth algebraic order multiderivative methods with phase-lag of order 12(2)18 is developed. Numerical application of the new obtained methods to the Schrodinger equation shows their efficiency compared with other similar well known methods of the literature. 

Simos30,31 has derived the first eighth algebraic order almost P-stable exponentially-fitted methods. The new methods have the form ... 

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... 

The coefficients of the method are defined in order to have the maximal algebraic order method and the minimal phase-lag. As a result an eighth-order method with phase-lag of order fourteen is produced. The interval of periodicity is equal to (0,8.35). 

We observe that the above family of methods contains free parameters w,. The local truncation error of the final families of methods is of order 0(/( ) (see ref. 14 for details), i.e. is of eighth algebraic order. 

Simos has considered the following eighth algebraic order explicit method ... 

In ref 154 the author presents an implicit hybrid two step method for the solution of second order initial value problem. The cost of the new obtained method is only six function evaluations per step and the algebraic order is eighth. The author studies the P-stability property and the conclusion is the new method satisfies this property requiring one stage less. 

In 40 the authors present a new explicit Runge-Kutta method with algebraic order four, minimum error of the fifth algebraic order (the limit of the error is zero, when the step-size tends to zero), infinite order of dispersion and eighth order of dissipation i.e. they present an optimized explicit Runge-Kutta method of fourth order. The efficiency of the newly developed method is shown through the numerical illustrations of a wide range of methods when these are applied to well-known periodic orbital problems. 

From the analysis the author concluded that the properties of the above two cases are the same i.e. they are both of algebraic order two, they have both phase-lag of order eighth and they have the same interval of periodicity (0,1). The author applied the new proposed methods to several well known problems of the literature (Duffing equation, two-body problem etc.) in order to show the efficiency of the new methods. 

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