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High-fidelity numerical solution of the time-dependent Dirac equation
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, Numerical Analysis.
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, Numerical Analysis.
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Inorganic Chemistry.
2014 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 262, 86-103 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
2014. Vol. 262, 86-103 p.
National Category
Computational Mathematics Theoretical Chemistry
Identifiers
URN: urn:nbn:se:uu:diva-215119DOI: 10.1016/j.jcp.2013.12.038ISI: 000330955200006OAI: oai:DiVA.org:uu-215119DiVA: diva2:686245
Available from: 2014-01-09 Created: 2014-01-10 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Efficient Simulation of Wave Phenomena
Open this publication in new window or tab >>Efficient Simulation of Wave Phenomena
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Wave phenomena appear in many fields of science such as acoustics, geophysics, and quantum mechanics. They can often be described by partial differential equations (PDEs). As PDEs typically are too difficult to solve by hand, the only option is to compute approximate solutions by implementing numerical methods on computers. Ideally, the numerical methods should produce accurate solutions at low computational cost. For wave propagation problems, high-order finite difference methods are known to be computationally cheap, but historically it has been difficult to construct stable methods. Thus, they have not been guaranteed to produce reasonable results.

In this thesis we consider finite difference methods on summation-by-parts (SBP) form. To impose boundary and interface conditions we use the simultaneous approximation term (SAT) method. The SBP-SAT technique is designed such that the numerical solution mimics the energy estimates satisfied by the true solution. Hence, SBP-SAT schemes are energy-stable by construction and guaranteed to converge to the true solution of well-posed linear PDE. The SBP-SAT framework provides a means to derive high-order methods without jeopardizing stability. Thus, they overcome most of the drawbacks historically associated with finite difference methods.

This thesis consists of three parts. The first part is devoted to improving existing SBP-SAT methods. In Papers I and II, we derive schemes with improved accuracy compared to standard schemes. In Paper III, we present an embedded boundary method that makes it easier to cope with complex geometries. The second part of the thesis shows how to apply the SBP-SAT method to wave propagation problems in acoustics (Paper IV) and quantum mechanics (Papers V and VI). The third part of the thesis, consisting of Paper VII, presents an efficient, fully explicit time-integration scheme well suited for locally refined meshes.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2017. 42 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1463
Keyword
finite difference method, high-order accuracy, stability, summation by parts, simultaneous approximation term, quantum mechanics, Dirac equation, local time-stepping
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-310124 (URN)978-91-554-9777-4 (ISBN)
Public defence
2017-02-10, Room 2446, ITC, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-01-19 Created: 2016-12-11 Last updated: 2017-01-25

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Almquist, MartinMattsson, KenEdvinsson, Tomas

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