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 semidiscretisation 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 generatedby differentiation and by multiplication with the interaction potential: it turns outthat this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaledby increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as O(M log M), where M is thenumber of degrees of freedom in the discretisation.
Original language  English 

Pages (fromto)  689720 
Number of pages  32 
Journal  Foundations of Computational Mathematics 
Volume  14 
Issue number  4 
Early online date  19 Feb 2014 
DOIs  
Publication status  Published  31 Aug 2014 
Keywords
 Exponential splittings
 Krylov subspace methods
 Semiclassical Schrödinger equation
 Spectral collocation
 Zassenhaus splitting
ASJC Scopus subject areas
 Analysis
 Computational Mathematics
 Computational Theory and Mathematics
 Applied Mathematics
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Pranav Singh
 Department of Mathematical Sciences  Senior Lecturer
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching