Convergence of the Magnus Series

Moan, P. C.; Niesen, Jitse

doi:10.1007/s10208-007-9010-0

Cited by 86 publications

(109 citation statements)

References 18 publications

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“…Here we state a theorem which, on the one hand, explains the phenomena observed by Moan and Niesen [27] and, on the other hand, provides a new tool to determine the actual convergence domain of the Magnus series in some physically relevant examples and applications.…”

Section: Another Characterization Of the Convergence Of The Magnus Sementioning

confidence: 86%

Sufficient conditions for the convergence of the Magnus expansion

Casas¹

2007

J. Phys. A: Math. Theor.

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Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equationThe first one provides a bound on the convergence domain based on the norm of the operator A(t). The second condition links the convergence of the expansion with the structure of the spectrum of Y (t), thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.

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Section: Another Characterization Of the Convergence Of The Magnus Sementioning

confidence: 86%

Sufficient conditions for the convergence of the Magnus expansion

Casas¹

2007

J. Phys. A: Math. Theor.

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“…Magnus [24] showed that the solution of the differential equation y = A(ξ)y can be written as y(ξ) = exp(Ω(ξ)) y(0), where the matrix Ω(ξ) is given by the infinite series [26] proved that the series converges if…”

Section: Magnus Methodsmentioning

confidence: 99%

Evaluating the Evans function: Order reduction in numerical methods

Malham

Niesen

2008

Math. Comp.

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Abstract. We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss-Legendre method has previously been used for such problems; however more recently, methods based on the Magnus expansion have been proposed. Here we extensively examine the stiff regime for a general scalar Schrödinger operator. We show that although the fourth-order Magnus method suffers from order reduction, a fortunate cancellation when computing the Evans matching function means that fourth-order convergence in the end result is preserved. The Gauss-Legendre method does not suffer from order reduction, but it does not experience the cancellation either, and thus it has the same order of convergence in the end result. Finally we discuss the relative merits of both methods as spectral tools.

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“…In general, the Magnus series does not converge unless A is small in a suitable sense. Several improved bounds to the actual radius of convergence in terms of A have been obtained along the years [18,1,15,17]. In this respect, the following result is proved in [6]: This theorem, in fact, provides the optimal convergence domain, in the sense that π is the largest constant for which the result holds without any further restrictions on the operator A(t).…”

Section: Magnus Series and Its Convergencementioning

confidence: 99%

New analytic approximations based on the Magnus expansion

2011

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The Magnus expansion is a frequently used tool to get approximate analytic solutions of time-dependent linear ordinary differential equations and in particular the Schrödinger equation in quantum mechanics. However, the complexity of the expansion restricts its use in practice only to the first terms. Here we introduce new and more accurate analytic approximations based on the Magnus expansion involving only univariate integrals which also shares with the exact solution its main qualitative and geometric properties.

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Convergence of the Magnus Series

Cited by 86 publications

References 18 publications

Sufficient conditions for the convergence of the Magnus expansion

Sufficient conditions for the convergence of the Magnus expansion

Evaluating the Evans function: Order reduction in numerical methods

New analytic approximations based on the Magnus expansion

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