Integral Equation Method for Coupled Schrödinger Equations

Gonzales, R. A.; Kang, Sheon-Young; Koltracht, I.; Rawitscher, George

doi:10.1006/jcph.1999.6272

Cited by 29 publications

(12 citation statements)

References 12 publications

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“…In the IEM, the boundary conditions are built in automatically via the Green's functions, while in the present form of the FEM the solutions with exponential growth are eliminated by forcing the closed channel component to be zero on the surface Σ. In a separate study of two coupled equations, 7 it is shown that there are situations in which the conventional Numerov method has severe difficulty in obtaining the correct asymptotic boundary condition, while both the IEM and the FEM do not.…”

Section: Discussionmentioning

confidence: 99%

“…There are various calculational methods available for dealing with this scattering problem: modified Numerov, Gordon's 3 , 4 a finite element method (FEM), 5 and a recently developed method that consist of replacing the coupled differential equations by equivalent integral equations (IEM), 6 . 7 In addition, there are more sophisticated finite difference methods 8 ; we, however, will not consider these methods here since they are not as widely in use. Methods involving the representation of a continuous function by a finite set of sampling points have been discussed.…”

Section: Introductionmentioning

confidence: 99%

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Comparison of numerical methods for the calculation of cold atom collisions

Rawitscher

Esry

Tiesinga

et al. 1999

The Journal of Chemical Physics

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Comparison between three different numerical techniques for solving a coupled channel Schrödinger equation is presented. The benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are i) a recently developed spectral type integral equation method based on Chebyshev expansions, ii) a finite element expansion, and iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods i) and ii), achieving an accuracy of ten significant figures with a double precision calculation. Method iii) achieves seven significant figures. The scattering length and effective range are also obtained.1

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparison of numerical methods for the calculation of cold atom collisions

Rawitscher

Esry

Tiesinga

et al. 1999

The Journal of Chemical Physics

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“…The restriction to spherically symmetric a has the advantage that the equations decouple for every degree and order of the spherical harmonics. That makes our method comparable to the corresponding integral equation method for the radial Schr€ odinger equation [4,16,17]. From there, and from other applications of the integral equation method [1,8,9,18,19,27], it is well known that the linear systems arising from the discretization of integral equations are generally well conditioned.…”

Section: Introductionmentioning

confidence: 89%

The integral equation method for a steady kinematic dynamo problem

Stefani

Gerbeth

2004

Journal of Computational Physics

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With only a few exceptions, the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method. However, the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics. The paper is intended to facilitate the use of integral equation solvers in dynamo theory. In concrete, the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with a spherically symmetric, isotropic helical turbulence parameter α. Three examples of the function α(r) with steady and oscillatory solutions are considered. A convergence rate proportional to the inverse squared of the number of grid points is achieved. Based on this method, a convergence accelerating strategy is developed and the convergence rate is improved remarkably. Typically, quite accurate results can be obtained with a few tens of grid points.

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“…Due to their importance in numerous applications, reaching from Bose-Einstein condensation over nonlinear optics up to plasma physics, nonlinear Schrödinger equations are nowadays very well studied numerically. In the last decades a large variety of different numerical schemes has been proposed [8,2,10,22,23]. Thanks to their simplicity and accuracy, a popular choice thereby lies in so-called splitting methods, where the right hand side of (1.1) is split into the linear and nonlinear part, respectively, see, e.g., [11,13,7] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Convergence, error analysis and longtime behavior of the scalar auxiliary variable method for the nonlinear Schrödinger equation

Poulain

Schratz

2021

IMA Journal of Numerical Analysis

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We carry out the convergence analysis of the scalar auxiliary variable (SAV) method applied to the nonlinear Schrödinger equation, which preserves a modified Hamiltonian on the discrete level. We derive a weak and strong convergence result, establish second-order global error bounds and present longtime error estimates on the modified Hamiltonian. In addition, we illustrate the favorable energy conservation of the SAV method compared to classical splitting schemes in certain applications.

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Integral Equation Method for Coupled Schrödinger Equations

Cited by 29 publications

References 12 publications

Comparison of numerical methods for the calculation of cold atom collisions

Comparison of numerical methods for the calculation of cold atom collisions

The integral equation method for a steady kinematic dynamo problem

Convergence, error analysis and longtime behavior of the scalar auxiliary variable method for the nonlinear Schrödinger equation

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