A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems

Runge‐Kutta (RK) pairs sharing orders 6 and 5 are widely used for solving initial value problems. Thus, we consider the dispersion from a standard cyclic orbit as a function of frequency v and present a modification of a particular pair. Namely, we derive a zero‐dissipative pair with vanished phase lag and its first derivative. This modified pair outperforms the other Runge‐Kutta 6(5) pairs as illustrated with various numerical tests in oscillatory problems.

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“…Another test problem was an inhomogeneous problem:

y^{''} = - 100 y (t) + 99 \sin (t), y (0) = 1, y^{'} (0) = 11,

with analytical solution,

y (t) = \cos (10 t) + \sin (10 t) + \sin (t) .

We integrated that problem in the interval

t \in ([], 0, 10 π)

using ω = 10 as in Alolyan et al…”

Section: Numerical Resultsmentioning

confidence: 99%

Fitted modifications of Runge‐Kutta pairs of orders 6(5)

Медведев

Simos

Tsitouras³

2018

Math Methods in App Sciences

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“…Four-step methods with variable coefficients have appeared already in the previous studies. [23][24][25] They attain only fourth algebraic order, and they do not use off-step points. Li and Wang 26 presented an extension of Runge-Kutta-Nyström type methods that was actually a special case of (2)(3)(4).…”

Section: Preliminariesmentioning

confidence: 99%

Hybrid, phase–fitted, four–step methods of seventh order for solving x″(t) = f(t,x)

Медведев

Simos

Tsitouras³

2019

Math Methods in App Sciences

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A four‐step method of seventh algebraic order is presented. It is tuned for addressing the special second order initial value problem. The new method is hybrid, explicit, and uses three stages per step. In addition is phase fitted. In consequence it uses variable coefficients that depend on the magnitude of the step‐size. We also present numerical tests on a set of standard problems that illustrate the efficiency of the derived method over older ones given in the relevant literature.

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“…The derivatives of the phase-lag for a 6-step symmetric nite difference method can be computed using the above mentioned formula (8) …”

Section: Remarkmentioning

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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A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems

Cited by 95 publications

References 27 publications

Fitted modifications of Runge‐Kutta pairs of orders 6(5)

Fitted modifications of Runge‐Kutta pairs of orders 6(5)

Hybrid, phase–fitted, four–step methods of seventh order for solving x″(t) = f(t,x)

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

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