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Numerical Methods for Wave Propagation: Analysis and Applications in Quantum Dynamics
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrödinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.

The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2016. , 33 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1330
Keyword [en]
computational wave propagation, quantum dynamics, time-dependent Schrödinger equation, spectral methods, Gaussian beams, splitting methods, low-rank approximation
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
URN: urn:nbn:se:uu:diva-268625ISBN: 978-91-554-9437-7 (print)OAI: oai:DiVA.org:uu-268625DiVA: diva2:878164
Public defence
2016-02-12, ITC 2446, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Projects
eSSENCE
Available from: 2016-01-19 Created: 2015-12-08 Last updated: 2016-02-12
List of papers
1. An adaptive pseudospectral method for wave packet dynamics
Open this publication in new window or tab >>An adaptive pseudospectral method for wave packet dynamics
2012 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 137, 044111:1-12 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-179058 (URN)10.1063/1.4737893 (DOI)000307611500012 ()
Projects
eSSENCE
Available from: 2012-07-26 Created: 2012-08-06 Last updated: 2017-12-07Bibliographically approved
2. Coupling of Gaussian beam and finite difference solvers for semiclassical Schrödinger equations
Open this publication in new window or tab >>Coupling of Gaussian beam and finite difference solvers for semiclassical Schrödinger equations
2015 (English)In: Advances in Applied Mathematics and Mechanics, ISSN 2070-0733, E-ISSN 2075-1354, Vol. 7, 687-714 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262240 (URN)10.4208/aamm.2013.m411 (DOI)000361055400001 ()
Projects
eSSENCE
Available from: 2015-09-09 Created: 2015-09-10 Last updated: 2017-12-04Bibliographically approved
3. Accelerated convergence for Schrödinger equations with non-smooth potentials
Open this publication in new window or tab >>Accelerated convergence for Schrödinger equations with non-smooth potentials
2014 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 54, 729-748 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-214685 (URN)10.1007/s10543-013-0465-x (DOI)000342210300009 ()
Projects
eSSENCE
Available from: 2014-01-01 Created: 2014-01-09 Last updated: 2017-12-06Bibliographically approved
4. Stiff convergence of force-gradient operator splitting methods
Open this publication in new window or tab >>Stiff convergence of force-gradient operator splitting methods
2015 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 94, 33-45 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-248024 (URN)10.1016/j.apnum.2015.03.005 (DOI)000355498600003 ()
Projects
eSSENCE
Available from: 2015-03-18 Created: 2015-03-26 Last updated: 2017-12-04Bibliographically approved
5. A matrix-free Legendre spectral method for initial–boundary value problems
Open this publication in new window or tab >>A matrix-free Legendre spectral method for initial–boundary value problems
2016 (English)In: Electronic Transactions on Numerical Analysis, ISSN 1068-9613, E-ISSN 1068-9613, Vol. 45, 283-304 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-267728 (URN)000393415700015 ()
Projects
eSSENCE
Available from: 2016-07-29 Created: 2015-11-25 Last updated: 2017-12-01Bibliographically approved
6. Discretized dynamical low-rank approximation in the presence of small singular values
Open this publication in new window or tab >>Discretized dynamical low-rank approximation in the presence of small singular values
2016 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, 1020-1038 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-267729 (URN)10.1137/15M1026791 (DOI)000375488100021 ()
Projects
eSSENCE
Available from: 2016-04-05 Created: 2015-11-25 Last updated: 2017-12-01Bibliographically approved

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