A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

Anastassi, Zacharias; Simos, T. E.

doi:10.1016/j.cam.2012.03.016

Cited by 136 publications

(32 citation statements)

References 37 publications

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“…For the numerical integration of the above mentioned initial value problem (1), one can use multistep methods of the form…”

Section: Analysis Of the Phase-lag For Symmetric Multistep Methodsmentioning

confidence: 99%

“…From the form (11) with k = 4 and b 4 = 0 we get the form of the explicit symmetric four-step methods [1]:…”

Section: The Family Of Explicit Four-step Methods With Vanished Phasementioning

confidence: 99%

“…In this paper the numerical solution of special second-order initial-value problems of the form y ′′ (x) = f (x, y), y(x 0 ) = y 0 and y ′ (x 0 ) = y ′ 0 (1) with an periodical and/or oscillatory solutions are investigated. The main characteristic of the mathematical models of the above mentioned problems is that the ordinary differential equations which describe the above models are of second order in which the first derivative y ′ does not appear explicitly (see for numerical methods for these problems [3] - [32] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…The main characteristic of the mathematical models of the above mentioned problems is that the ordinary differential equations which describe the above models are of second order in which the first derivative y ′ does not appear explicitly (see for numerical methods for these problems [3] - [32] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative

Simos¹

2014

Appl. Math. Inf. Sci.

121

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Abstract:In the present paper, we will investigate a family of explicit four-step methods first introduced by Anastassi and Simos [1] for the case of vanishing of phase-lag and its first derivative. These methods are efficient for the numerical solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions. As we mentioned before the main scope of this paper is the study of the elimination of the phase-lag and its first derivative of the family of the methods to be investigated. A comparative error and stability analysis will be presented for the studied family of four-step explicit methods. The new obtained methods will finally be tested on the resonance problem of the Schrödinger equation in order to examine their efficiency.

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“…For the numerical integration of the above mentioned initial value problem (1), one can use multistep methods of the form…”

Section: Analysis Of the Phase-lag For Symmetric Multistep Methodsmentioning

confidence: 99%

“…From the form (11) with k = 4 and b 4 = 0 we get the form of the explicit symmetric four-step methods [1]:…”

Section: The Family Of Explicit Four-step Methods With Vanished Phasementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative

Simos¹

2014

Appl. Math. Inf. Sci.

121

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Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs

Simos

Tsitouras²

2017

Math Methods in App Sciences

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In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.

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A new family of 7 stages, eighth‐order explicit Numerov‐type methods

Simos

Tsitouras²

2017

Math Methods in App Sciences

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In this paper, we consider the integration of the special second-order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase-lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given. KEYWORDSconstant coefficients, hybrid Numerov methods, initial value problem, interval of periodicity, numerical solution, phase-lag INTRODUCTIONWe consider the initial value problem of second-orderwhere f ∶ ℜ N+1 → ℜ N and y [0] , y ′[0] ∈ ℜ N . Equation 1 is of special form since the derivative y ′ is not included in f.Here, we will deal with numerical methods solving problem (1) especially when their solution is oscillatory. Many problems in theoretical physics and chemistry, in electronics, in celestial mechanics, quantum mechanical scattering theory, and many fields of engineering can be solved applying such methods. P-stability is a very useful property when dealing with periodic problems, 1,2 and implicit Numerov-type methods with off-step points were introduced by Hairer, 3 Cash, 4 and Chawla 5 especially for achieving this property.Phase-lag is another interesting property introduced by Brusa and Nigro. 6 In this case, the numerical solutions stay as close as possible to the angle of the theoretical solution of the simple harmonic oscillator, which serves as test problem. Explicit modifications of Numerov methods with reduced phase errors were constructed later by Chawla 7 and Chawla and Rao. [8][9][10] Math Meth Appl Sci. 2017;40:7867-7878.wileyonlinelibrary.com/journal/mma

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A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

Cited by 136 publications

References 37 publications

On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative

On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative

Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs

A new family of 7 stages, eighth‐order explicit Numerov‐type methods

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