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Foundations of Computational Mathematics
Published in: Foundations of Computational Mathematics 4/2014

01-08-2014

Effective Approximation for the Semiclassical Schrödinger Equation

Authors: Philipp Bader, Arieh Iserles, Karolina Kropielnicka, Pranav Singh

Published in: Foundations of Computational Mathematics | Issue 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.
previous article On the Numerical Stability of Fourier Extensions
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Footnotes
1
Unless a non-existing term is subtracted and thus newly introduced instead of removed.
 
2
Using a Fourier basis the cost is \({\mathcal {O}}\!\left( M\log M\right) \).
 
3
As before, a tilde denotes a discretisation.
 
4
All powers of \(\tau \) are odd because of the palindromy of the symmetric BCH formula. Since \(\tau ={\mathrm {i}}\Delta t\), this means that they always contribute a multiple of \(\pm {\mathrm {i}}\).
 
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Metadata
Title
Effective Approximation for the Semiclassical Schrödinger Equation
Authors
Philipp Bader
Arieh Iserles
Karolina Kropielnicka
Pranav Singh
Publication date
01-08-2014
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 4/2014
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-013-9182-8

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