CP methods of higher order for Sturm–Liouville and Schrödinger equations

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

doi:10.1016/j.cpc.2004.07.001

Cited by 41 publications

(56 citation statements)

References 11 publications

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“…In [35] a Magnus scheme of order 10 is described where ν = 5 and in [32] PPM-schemes up to order 16 are presented. Note that only the terms where the degree in h is smaller or equal to the required degree of the method have to be included in the algorithm, for instance in the approximation of σ 3 and σ 4 the term inQ 3 4 can be disregarded in the Magnus scheme of order 10.…”

Section: Quadrature Of the (Multivariate) Integralsmentioning

confidence: 99%

“…To approximate the integrals in (26) quadrature must be used which can deal adequately with the oscillatory entries of the matrix function B. In section 4.3 a Filon-type quadrature rule will be discussed which is very similar to the procedure used in the description of high order PPMs in [28,32]. There the potential function q is replaced by a piecewise polynomial, which makes the integrals in (26) analytically solvable.…”

Section: Modified Neumann and Magnus Schemes For The Schrödinger Equamentioning

confidence: 99%

“…An alternative way to apply the Filon-type rule is by approximating q(x) (piecewisely) by a series over shifted Legendre polynomials (as in done in the description of PPM [28,32]):…”

Section: Quadrature Of the (Multivariate) Integralsmentioning

confidence: 99%

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Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Ledoux

Daele

Berghe

2009

Computer Physics Communications

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Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the SturmLiouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods.

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Section: Quadrature Of the (Multivariate) Integralsmentioning

confidence: 99%

Section: Modified Neumann and Magnus Schemes For The Schrödinger Equamentioning

confidence: 99%

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Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Ledoux

Daele

Berghe

2009

Computer Physics Communications

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“…In [7] and [11] some higher order CP versions, the so-called CPM{P, N } methods were found to be well suited for the solution of the one-dimensional Schrödinger problem. More recently these CPM{P, N } formulae were generalised to the coupled channel case (see [12]).…”

Section: The Cp Algorithm For the Multichannel Casementioning

confidence: 99%

Piecewise Constant Perturbation Methods for the Multichannel Schrödinger Equation

Ledoux

Daele

Berghe

2006

Computational Science – ICCS 2006

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Abstract. The CPM{P, N} methods form a class of methods specially devised for the propagation of the solution of the one-dimensional Schrö-dinger equation. Using these CPM{P, N} methods in a shooting procedure, eigenvalues of the boundary value problem are obtained to very high precision. Some recent advances allowed the generalization of the CPM{P, N} methods to systems of coupled Schrödinger equations. Also for these generalised CPM{P, N} methods a shooting procedure can be formulated, solving the multichannel bound state problem.

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“…The so-called piecewise perturbation methods (PPM) use a perturbation technique to construct some correction terms that are added to the known solution of the approximating problem with a piecewise-constant potential. In this way methods up to order 16 were constructed (see [13,15,17]) which can efficiently compute the eigenvalues of regular Schrödinger problems y ′′ (x) = (V (x) − E)y(x).…”

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confidence: 99%

Solution of Sturm–Liouville Problems Using Modified Neumann Schemes

Ledoux¹,

Daele²

2010

SIAM J. Sci. Comput.

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Abstract. The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrödinger problems to more general Sturm-Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method, but provides for higher order approximations. We show that the method can be successfully implemented in a competitive automatic general-purpose software package.

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CP methods of higher order for Sturm–Liouville and Schrödinger equations

Cited by 41 publications

References 11 publications

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Piecewise Constant Perturbation Methods for the Multichannel Schrödinger Equation

Solution of Sturm–Liouville Problems Using Modified Neumann Schemes

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