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Finite Difference and Discontinuous Galerkin Methods for Wave Equations
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.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost.

There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2017. , 53 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1522
Keyword [en]
Wave propagation, Finite difference method, Discontinuous Galerkin method, Stability, Accuracy, Summation by parts, Normal mode analysis
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-320614ISBN: 978-91-554-9927-3 (print)OAI: oai:DiVA.org:uu-320614DiVA: diva2:1090174
Public defence
2017-06-13, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-05-22 Created: 2017-04-23 Last updated: 2017-06-28
List of papers
1. High order finite difference methods for the wave equation with non-conforming grid interfaces
Open this publication in new window or tab >>High order finite difference methods for the wave equation with non-conforming grid interfaces
2016 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 68, 1002-1028 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-264754 (URN)10.1007/s10915-016-0165-1 (DOI)000380693700006 ()
External cooperation:
Available from: 2016-01-27 Created: 2015-10-16 Last updated: 2017-04-23Bibliographically approved
2. An improved high order finite difference method for non-conforming grid interfaces for the wave equation
Open this publication in new window or tab >>An improved high order finite difference method for non-conforming grid interfaces for the wave equation
2017 (English)In: Computing Research Repository, no 1702.02056Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-320600 (URN)
Available from: 2017-02-07 Created: 2017-04-23 Last updated: 2017-06-30Bibliographically approved
3. Convergence of summation-by-parts finite difference methods for the wave equation
Open this publication in new window or tab >>Convergence of summation-by-parts finite difference methods for the wave equation
2017 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 71, 219-245 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-264752 (URN)10.1007/s10915-016-0297-3 (DOI)000398062500009 ()
Available from: 2016-09-27 Created: 2015-10-16 Last updated: 2017-05-17Bibliographically approved
4. Convergence of finite difference methods for the wave equation in two space dimensions
Open this publication in new window or tab >>Convergence of finite difference methods for the wave equation in two space dimensions
2017 (English)In: Computing Research Repository, no 1702.01383Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-320603 (URN)
Available from: 2017-05-18 Created: 2017-04-23 Last updated: 2017-06-30Bibliographically approved
5. An energy based discontinuous Galerkin method for acoustic–elastic waves
Open this publication in new window or tab >>An energy based discontinuous Galerkin method for acoustic–elastic waves
2017 (English)In: Proc. 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Minneapolis, MN: University of Minnesota Press, 2017, 389-390 p.Conference paper, Abstract (Other academic)
Place, publisher, year, edition, pages
Minneapolis, MN: University of Minnesota Press, 2017
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-320601 (URN)
Conference
WAVES 2017
Available from: 2017-05-19 Created: 2017-04-23 Last updated: 2017-06-30Bibliographically approved

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