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A four stages twelfth algebraic order symmetric two-step method with vanished phase-lag and its first and second derivative is developed in this paper for the first time in the literature. More specifically in this paper the following will be investigated: (i) the phase-lag analysis for the development of the new method, (ii) the construction of the new method, (iii) the local truncation error analysis which is based on the radial Schrödinger equation, (iv) the interval of periodicity analysis and the stability analysis which is based on a scalar test equation with frequency not equal with the frequency of the scalar test equation used for the phase-lag analysis, (v) the error estimation procedure based on the algebraic order, (vi) the numerical results from our numerical tests for the study of the efficiency of the new obtained method. The numerical tests are based on the numerical solution of the Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper we introduce, for the first time in the literature, an optimized multistage symmetric two-step method. This method is considered as optimized due to the following reasons: (1) it is of tenth-algebraic order scheme, (2) it has obliterated the phase-lag and its first, second, third and fourth derivatives, (3) it has improved stability characteristics, (4) it is a P-stable method. For the new proposed multistage symmetric two-step method we present a full theoretical investigation consisted of: (1) local truncation error and comparative error analysis, (2) stability analysis and (3) interval of periodicity analysis. The effectiveness of the new builded multistage symmetric two-step method is evaluated on the solution of systems of coupled differential equations of the Schrödinger type. [ABSTRACT FROM AUTHOR]

In this paper and for the first time in the literature we develop a new three-stages symmetric two-step method with improved properties. More specifically the new scheme: (1) is of symmetric type, (2) is of two-step algorithm, (3) is of three-stages, (4) it is of tenth-algebraic order, (5) it has eliminated the phase-lag and its first derivative, (6) it has improved stability properties for the general problems, (7) it is a P-stable method since it has an interval of periodicity equal to 0,∞. In order to develop the new hybrid algorithm we use the following approximations: (i) An approximation developed on the first layer on the point xn-1, (ii) An approximation developed on the second layer on the point xn and finally, (iii) An approximation developed on the third (final) layer on the point xn+1. For the new proposed method we give a full theoretical analysis (local truncation error analysis and stability and interval of periodicity analysis). The efficiency of the new proposed method is examined on the numerical solution of coupled differential equations arising from the Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper and for the first time in the literature, we build a new hybrid symmetric two-step method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of two-step, (3) the new scheme is of five-stages, (4) the new scheme is of twelfth-algebraic order, (5) the new scheme has eliminated the phase-lag and its first, second, third, fourth and fifth derivatives, (6) the new scheme has improved stability characteristics for the general problems, (7) the new scheme is P-stable [with interval of periodicity equal to 0,∞] and (8) the new scheme builded based on the following approximations:the first stage is approximation on the point xn-1,the second stage is approximation on the point xn-1,the third stage is approximation on the point xn-1,the fourth stage is approximation on the point xn and finally,the fifth stage is approximation on the point xn+1, For the new builded scheme we give a full numerical analysis (local truncation error and stability analysis). The efficiency of the new builded scheme is examined with the numerical solution of systems of coupled differential equations of the Schrödinger type. [ABSTRACT FROM AUTHOR]