A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation

Alolyan, Ibraheem; Simos, T. E.

doi:10.1007/s10910-013-0301-1

Cited by 80 publications

(4 citation statements)

References 97 publications

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“…6. The Method developed in [58], which is indicated as Method HYBPLDDDEA is more efficient than method developed in [45], which is indicated as Method RKTPLDDEA. 7.…”

Section: Remark 17mentioning

confidence: 99%

“…9 Accuracy (digits) for several values of C PU time (in seconds) for the eigenvalue E 2 = 341.495874. The nonexistence of a value of accuracy (digits) indicates that for this value of CPU, accuracy (digits) is less than 0 -The Phase-Fitted Method (Case 2) developed in [48], which is indicated as Method NMPF2 -The Method developed in [52] (Case 2), which is indicated as Method NMC2 -The Method developed in [52] (Case 1), which is indicated as Method NMC1 -The Method developed in [45], which is indicated as Method RKTPLDDEA -The Method developed in [58], which is indicated as Method HYBPLDDDEA -The Hybrid Low Computational Computational Cost Four-Step Method developed in [46], which is indicated as Method HYMETH8 -The New Obtained Three Stages Explicit Symmetric Four-Step Method which is developed in Sect. 5, which is indicated as Method MuSMeth10…”

Section: Remark 16mentioning

confidence: 99%

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A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

Alolyan

Simos

2015

J Math Chem

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A new Multistage high algebraic order four-step method is obtained in this paper. It is the first time in the literature that a method of this category is developed and has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives. We study this new method by investigating: (1) the development of the new method, i.e. the calculation of the coefficients of the method in order the phase-lag and its first, second, third, fourth and fifth derivatives of the phase-lag to be vanished, (2) the determination of the formula of the Local Truncation Error, (3) the comparative analysis of the Local Truncation Error (with this we mean the application of the new method and similar methods on a test problem and the analysis of their behavior), (4) the stability of the new method, by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the results of this application i.e. by investigating the interval of periodicity of the new obtained method. We finally study the computational behavior the new Electronic supplementary material The online version of this article (123 J Math Chem developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrödinger equation. We prove the effectiveness of the new obtained method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

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“…6. The Method developed in [58], which is indicated as Method HYBPLDDDEA is more efficient than method developed in [45], which is indicated as Method RKTPLDDEA. 7.…”

Section: Remark 17mentioning

confidence: 99%

Section: Remark 16mentioning

confidence: 99%

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

Alolyan

Simos

2015

J Math Chem

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“…the local truncation error analysis. In this research, we will compare the results of our new produced methods with the analysis of the local truncation error of other methods of the same form (comparative local truncation error analysis), 2. the stability analysis of the resulting methods using scalar test equation with frequency different than the scalar test equation for the phase-lag analysis 3. the numerical experiments produced by the application of the new obtained methods to the resonance problem of the one dimensional time independent Schrödinger equation (see for more details [5]). …”

Section: Remarkmentioning

confidence: 99%

“…The symmetric 2 m-step method with characteristic equation given by (5) has phase-lag order q and phase-lag constant c given by:…”

Section: Theorem 1 [4]mentioning

confidence: 99%

Implicit three-step methods with phase-lag and its derivatives equal to zero for the numerical solution of the Schrödinger equation and related problems

Alolyan¹,

Simos²

2015

AIP Conference Proceedings

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Articles you may be interested inA linear four-step method with vanished phase-lag and its first and second derivatives for the numerical solution of periodic initial and/or boundary value problems AIP Conf. Proc. 1618, 1031 (2014); 10.1063/1.4897910 Six-step methods with vanished phase-lag and its first derivatives for the approximate solution of the Schrödinger equation and related problems AIP Conf. Proc. 1618, 1026 (2014); 10.1063/1.4897909Four-step hybrid type methods with vanished phase-lag and its derivatives for each level for the numerical solution of the Schrödinger equation and related problems AIP Conf. Abstract.A family of implicit three-step methods for the numerical solution of special second order periodic initial or boundary-value problems (without rst derivative in their mathematical model) is investigated in this paper. The construction of the methods of the present family is based on the request phase-lag and its derivatives to be equal to zero. The study of the methods of the present family included • the investigation on the local truncation error of the methods and its comparison with the local truncation error of the corresponding methods with constant coef cients • the investigation on the stability for the new produced methods (using scalar test equation with frequency different than the scalar test equation for the phase-lag analysis) • the application of the new constructed methods to the resonance problem of the radial time independent Schrödinger equation in order to show their ef ciency.

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An explicit linear six-step method with vanished phase-lag and its first derivative

Simos

2014

J Math Chem

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A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation

Cited by 80 publications

References 97 publications

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

Implicit three-step methods with phase-lag and its derivatives equal to zero for the numerical solution of the Schrödinger equation and related problems

An explicit linear six-step method with vanished phase-lag and its first derivative

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