uu.seUppsala University Publications

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Efficient Simulation of Wave PhenomenaPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis, 2017. , p. 42
##### Series

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1463
##### Keywords [en]

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: urn:nbn:se:uu:diva-310124ISBN: 978-91-554-9777-4 (print)OAI: oai:DiVA.org:uu-310124DiVA, id: diva2:1055093
##### Public defence

2017-02-10, Room 2446, ITC, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt1130",{id:"formSmash:j_idt1130",widgetVar:"widget_formSmash_j_idt1130",multiple:true}); Available from: 2017-01-19 Created: 2016-12-11 Last updated: 2017-01-25
##### List of papers

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.

1. A solution to the stability issues with block norm summation by parts operators$(function(){PrimeFaces.cw("OverlayPanel","overlay641392",{id:"formSmash:j_idt1179:0:j_idt1183",widgetVar:"overlay641392",target:"formSmash:j_idt1179:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Optimal diagonal-norm SBP operators$(function(){PrimeFaces.cw("OverlayPanel","overlay688425",{id:"formSmash:j_idt1179:1:j_idt1183",widgetVar:"overlay688425",target:"formSmash:j_idt1179:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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5. High-fidelity numerical solution of the time-dependent Dirac equation$(function(){PrimeFaces.cw("OverlayPanel","overlay686245",{id:"formSmash:j_idt1179:4:j_idt1183",widgetVar:"overlay686245",target:"formSmash:j_idt1179:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Realization of adiabatic Aharonov–Bohm scattering with neutrons$(function(){PrimeFaces.cw("OverlayPanel","overlay793968",{id:"formSmash:j_idt1179:5:j_idt1183",widgetVar:"overlay793968",target:"formSmash:j_idt1179:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

7. Multilevel local time-stepping methods of Runge–Kutta-type for wave equations$(function(){PrimeFaces.cw("OverlayPanel","overlay1055082",{id:"formSmash:j_idt1179:6:j_idt1183",widgetVar:"overlay1055082",target:"formSmash:j_idt1179:6:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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