Methods with Non-Empty Interval of Periodicity

2 Methods with Non Empty Interval of Periodicity. - 4.1.2.1 Fourth Algebraic Order Methods. - Simos and Mousadis,71 Simos and Tougelidis72 and Simos73 have considered the following family of methods  

The value of parameters a, b and c i = 1(1)3 are determined in order to obtain the method with minimal phase-lag. For the method of Simos and Mousadis71 ci — C2 = 0 and the other coefficients are determined in order to produce a method with phase-lag of order ten and with an interval of periodicity equal to (0, /720). For the method of Simos and Tougelidis72 c = 0 and the other coefficients are determined in order to produce a method with phase-lag of order twelve and with an interval of periodicity equal to (0,2(2772) ). This method is an improvement of the method produced by Simos and Mousadis.71 Finally, for the method Simos73 the coefficients are determined in order to produce a method with phase-lag of order fourteen and with an interval of periodicity equal to (0,29.0974). This method is an improvement of the method produced by Simos and Tougelidis.72 

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

For fl, = 0, / = 4(—1)0 the method obtained is the method of Allison, Raptis and Simos,75 which was the first hybrid method of algebraic order eight in the literature. The method has an interval of periodicity equal to [0,12.2774]. Jamieson76 makes some fruitful comments on the above formula concerning specific cases of the potential. 

If the free parameters a i = 4(—1)0 are defined in order to minimize the phase-lag then we have the method obtained by Simos.77 We have the following cases  

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In Table 8 we present the properties of the methods with non-empty interval of periodicity described above. 

Table 8 Properties of numerical methods with non-empty interval of periodicity. 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. I/E is an indication for implicit or explicit methods...

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