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Foundations of Computational Mathematics

Erschienen in:

01.08.2014

Effective Approximation for the Semiclassical Schrödinger Equation

verfasst von: Philipp Bader, Arieh Iserles, Karolina Kropielnicka, Pranav Singh

Erschienen in: Foundations of Computational Mathematics | Ausgabe 4/2014

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Abstract

The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences with the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential: it turns out that this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as \({\mathcal {O}}\!\left( M\log M\right) \), where \(M\) is the number of degrees of freedom in the discretisation.

Vorheriger Artikel On the Numerical Stability of Fourier Extensions

Nächster Artikel Computing Cup Products in -Cohomology of 3D Polyhedral Complexes

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Unless a non-existing term is subtracted and thus newly introduced instead of removed.

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Titel: Effective Approximation for the Semiclassical Schrödinger Equation
verfasst von: Philipp Bader
Arieh Iserles
Karolina Kropielnicka
Pranav Singh
Publikationsdatum: 01.08.2014
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 4/2014
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-013-9182-8

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