Efficient numerical solution of the one-dimensional Schrodinger eigenvalue problem using Magnus integrators

Ledoux, Veerle; Daele, Maarten Van; Berghe, Guido Vanden

doi:10.1093/imanum/drn062

Cited by 10 publications

(33 citation statements)

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“…We note that the Fer streamers approximations and the methods in [2,9] preserve this geometric property, unlike the methods in [7].…”

Section: Geometric Integrationmentioning

confidence: 94%

“…e.g., in the piecewise perturbation methods [7], in the right-correction Magnus series [2] and in the modified Magnus methods [9]. In addition, one can consider another regime.…”

Section: Uniform Versus Asymptotic Error Estimatesmentioning

confidence: 99%

“…In addition, quadrature estimates under λ,h 1 are often valid only in the absence of critical points and subject to a non-resonance condition [6]. In the context of the integral series in [2,7,9,14] and this paper, these considerations are key since the non-resonance condition is not satisfied in the integral (which appears in higher order approximations)…”

Section: Uniform Versus Asymptotic Error Estimatesmentioning

confidence: 99%

“…Since [2, p. 423] assumes that λ q max , this issue does not arise. Ledoux et al [9] does not assume λ ≥ q min , and disregards this issue. As discussed in [14], there are Sturm-Liouville problems where Eq.…”

Section: Number Of Steps In Each Numerical Meshmentioning

confidence: 99%

“…In [2] the knowledge of q max is required since it focuses on λ q max . Ledoux et al [9] does not use this information, since it does not control the positive argument of certain hyperbolic functions for λ < q min .…”

Section: Number Of Steps In Each Numerical Meshmentioning

confidence: 99%

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Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers

Ramos

2017

Adv Comput Math

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The current paper concerns the uniform and high-order discretization of the novel approach to the computation of Sturm-Liouville problems via Fer streamers, put forth in Ramos and Iserles (Numer. Math., 131(3), 2015). In particular, the discretization schemes are shown to enjoy large step sizes uniform over the entire eigenvalue range and tight error estimates uniform for every eigenvalue. They are made explicit for global orders 4,7,10. In addition, the present paper provides total error estimates that quantify the interplay between the truncation and the discretization in the approach by Fer streamers.

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“…We note that the Fer streamers approximations and the methods in [2,9] preserve this geometric property, unlike the methods in [7].…”

Section: Geometric Integrationmentioning

confidence: 94%

“…e.g., in the piecewise perturbation methods [7], in the right-correction Magnus series [2] and in the modified Magnus methods [9]. In addition, one can consider another regime.…”

Section: Uniform Versus Asymptotic Error Estimatesmentioning

confidence: 99%

Section: Uniform Versus Asymptotic Error Estimatesmentioning

confidence: 99%

Section: Number Of Steps In Each Numerical Meshmentioning

confidence: 99%

Section: Number Of Steps In Each Numerical Meshmentioning

confidence: 99%

See 3 more Smart Citations

Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers

Ramos

2017

Adv Comput Math

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Using a (Higher-Order) Magnus Method to Solve the Sturm-Liouville Problem

Ledoux

Daele

Berghe

2008

Computational Science – ICCS 2008

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Abstract. The main purpose of this paper is to describe techniques for the numerical solution of a Sturm-Liouville equation (in its Schrödinger form) by employing a Magnus expansion. With a suitable method to approximate the highly oscillatory integrals which appear in the Magnus series, high order schemes can be constructed. A method of order ten is presented. Even when the solution is highly-oscillatory, the scheme can accurately integrate the problem using stepsizes typically much larger than the solution "wavelength". This makes the method well suited to be applied in a shooting process to locate the eigenvalues of a boundary value problem.

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On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation

Ledoux

Daele

2011

Z. Angew. Math. Phys.

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Abstract. The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schrödinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru [21], this problem can be circumvented by using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher order polynomials, since these methods are equivalent to the CPM. Also LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.Mathematics Subject Classification (2010). 65L15; 65Z05; 34B09; 34L16.

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Efficient numerical solution of the one-dimensional Schrodinger eigenvalue problem using Magnus integrators

Cited by 10 publications

References 0 publications

Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers

Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers

Using a (Higher-Order) Magnus Method to Solve the Sturm-Liouville Problem

On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation

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