A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation

Пузынин, И.В.; Selin, A.V.; Vinitsky, S. I.

doi:10.1016/s0010-4655(99)00224-6

Cited by 28 publications

(25 citation statements)

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“…We have found these very useful for making long-time or large-space problems tractable [7], but we will not discuss such boundary conditions further in this paper. Puzynin et al [19,20] indicate how to generalize the time development to higher order, but do not discuss spatial integration.…”

Section: Introductionmentioning

confidence: 99%

Accurate numerical solutions of the time-dependent Schrödinger equation

Dijk

Toyama

2007

Phys. Rev. E

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We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schrödinger equation. The generalization yields numerical solutions accurate to order (∆x) 2r−1 in space and (∆t) 2M in time for any positive integers r and M , while CN employ r = M = 1. We note dramatic improvement in the attainable precision (circa 10 or greater orders of magnitude) along with several orders of magnitude reduction of computational time. The improved method is shown to lead to feasible studies of coherent-state oscillations with additional short-range interactions, wavepacket scattering, and long-time studies of decaying systems.

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Section: Introductionmentioning

confidence: 99%

Accurate numerical solutions of the time-dependent Schrödinger equation

Dijk

Toyama

2007

Phys. Rev. E

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“…There are well known problems with the computation of the coefficient b(ζ k ). We used the bi-directional algorithm [17] to find b(ζ k ) by the formula (11). Figure 5 presents the errors (68) of computing the phase coefficients for the maximum eigenvalue ζ 0 (61) with respect to the amplitude A of the potential (59).…”

Section: Numerical Results For Discrete Spectrummentioning

confidence: 99%

“…For t = τ formulas (46) coincide with (44)-(45). For the Schrödinger equation with a time-dependent operator, decomposition was obtained in a series of papers [11,12], but the authors decomposed the matrix Q in the Galerkin series and did not use difference schemes to find derivatives of Q.…”

Section: Magnus Expansionmentioning

confidence: 99%

Exponential fourth order schemes for direct Zakharov-Shabat problem

et al. 2019

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“…A comparison of some of these methods is given in [98]. More recently, Puzynin et aI., [122] used a FEG approach in space with high order time discretizations arising from diagonal Pad6 approximations.…”

Section: Multidimensional Case 221 Linear Problemsmentioning

confidence: 99%

Numerical Methods for Schrödinger-Type Problems

Fairweather¹,

Khebchareon²

2002

Applied Optimization

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Several physical phenomena are modeled by initial-boundary value problems which can be formulated as SchrOdinger -type systems of partial differential equations. In this paper, two classes of problems of this kind, Schrodinger equations, which arise in various areas of physics, and certain vibration problems from civil and mechanical engineering, are considered. A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrodinger equation, and to fourth order parabolic equations arising in vibrating beam and plate problems. Recently developed finite element methods for solving SchrOdinger-type systems are also outlined.Keywords: Schrodinger systems, linear and nonlinear Schrodinger equations, parabolic wave equation, vibrating beams and plates, fourth order parabolic equations, finite difference methods, finite element Galerkin methods, orthogonal spline collocation methods, spectral methods IntroductionWe consider the numerical solution of problems that can be written as initialboundary value problems (IBVPs) for real systems of partial differential equations of the form

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A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation

Cited by 28 publications

References 0 publications

Accurate numerical solutions of the time-dependent Schrödinger equation

Accurate numerical solutions of the time-dependent Schrödinger equation

Exponential fourth order schemes for direct Zakharov-Shabat problem

Numerical Methods for Schrödinger-Type Problems

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