Numerical Methods for Wave Propagation: Analysis and Applications in Quantum Dynamics
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1330
computational wave propagation, quantum dynamics, time-dependent Schrödinger equation, spectral methods, Gaussian beams, splitting methods, low-rank approximation
Research subject Scientific Computing
IdentifiersURN: urn:nbn:se:uu:diva-268625ISBN: 978-91-554-9437-7OAI: oai:DiVA.org:uu-268625DiVA: diva2:878164
2016-02-12, ITC 2446, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Holmgren, SverkerGradinaru, VasileKarlsson, Hans O.
List of papers