Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems

Kosti, A. A.; Anastassi, Zacharias; Simos, T. E.

doi:10.1016/j.camwa.2011.04.046

Cited by 138 publications

(30 citation statements)

References 9 publications

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“…-Phase-fitted methods and numerical methods with minimal phase-lag of RungeKutta and Runge-Kutta Nyström type have been obtained in [5][6][7][8][9][10][11][12][13][14]. -In [15][16][17][18][19][20], exponentially and trigonometrically fitted Runge-Kutta and RungeKutta Nyström methods are constructed.…”

Section: Recent Literature On the Subject Of The Papermentioning

confidence: 99%

“…6 is the basis of the interval of periodicity analysis for the new three stage explicit symmetric four-step method (10) with the coefficients (11) and (19).…”

Section: Stability Analysismentioning

confidence: 99%

“…7 s-v plane of the new three stage explicit symmetric four-step (10) with the coefficients given by (11) and (19) Definition 2 (see [21]) If for a method its interval of periodicity is equal to (0, ∞), then this method is called P-stable.…”

Section: Remark 11mentioning

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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Section: Recent Literature On the Subject Of The Papermentioning

confidence: 99%

“…6 is the basis of the interval of periodicity analysis for the new three stage explicit symmetric four-step method (10) with the coefficients (11) and (19).…”

Section: Stability Analysismentioning

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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“…Phase-fitted methods and numerical methods with minimal phase-lag of RungeKutta and Runge-Kutta Nyström type have been developed in [5][6][7][8][9][10][11][12][13][14]. 2.…”

Section: Remarkmentioning

confidence: 99%

A new eight algebraic order embedded explicit six-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation

Alolyan

Simos

2016

J Math Chem

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A new embedded symmetric six-step method with vanished phase-lag and its first, second, third and fourth derivatives is obtained for the first time in the literature in this paper. More precisely, for this new method we will investigate: -The development of the methods, i.e., the definition of the coefficients of the new method in order the phase-lag and its first, second, third and fourth derivatives of the phase-lag to be vanished, -The computation of the formulae of the Local Truncation Error (LTE) of the methods of the new family of methods, -The computation of the asymptotic forms of the LTE by applying of the new developed method on a scalar test problem (which is the time independent radial Schrödinger equation), -The computation of the asymptotic forms of the LTE of other methods of the same family and the comparative local truncation error analysis of all the methods of the family, -The stability (interval of periodicity) analysis of the new obtained method. To study the stability, we will use a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis.123 J Math Chem -The behavior of the new produced method by applying them to the numerical solution of the resonance problem of the radial Schrödinger equation. We will prove its efficiency by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

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“…q ′′ (x) = f (x, q), q(x 0 ) = q 0 and q ′ (x 0 ) = q ′ 0 (1) which have solutions with periodical and/or oscillatory behavior.…”

Section: Introductionmentioning

confidence: 98%

High algebraic order Runge–Kutta type two-step method with vanished phase-lag and its first, second, third, fourth, fifth and sixth derivatives

Ning

Simos

2015

Computer Physics Communications

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Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems

Cited by 138 publications

References 9 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

A new eight algebraic order embedded explicit six-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation

High algebraic order Runge–Kutta type two-step method with vanished phase-lag and its first, second, third, fourth, fifth and sixth derivatives

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