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| Content Provider | Springer Nature Link |
|---|---|
| Author | Iserles, Arieh Singh, Pranav Bader, Philipp Kropielnicka, Karolina |
| Copyright Year | 2014 |
| 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. |
| Ending Page | 720 |
| Page Count | 32 |
| Starting Page | 689 |
| File Format | |
| ISSN | 16153375 |
| e-ISSN | 16153383 |
| Journal | Foundations of Computational Mathematics |
| Issue Number | 4 |
| Volume Number | 14 |
| Language | English |
| Publisher | Springer US |
| Publisher Date | 2014-02-19 |
| Publisher Place | Boston |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Matrix exponential and similar matrix functions Economics general Semiclassical Schrödinger equation Krylov subspace methods Exponential splittings Linear and Multilinear Algebras, Matrix Theory Zassenhaus splitting Spectral, collocation and related methods Spectral collocation Numerical Analysis Computer Science Initial value problems Applications of Mathematics Math Applications in Computer Science Time-dependent Schrödinger equations, Dirac equations |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics |
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