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Convergence of summation-by-parts finite difference methods for the wave 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.
2017 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 71, 219-245 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
2017. Vol. 71, 219-245 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-264752DOI: 10.1007/s10915-016-0297-3ISI: 000398062500009OAI: oai:DiVA.org:uu-264752DiVA: diva2:861401
Available from: 2016-09-27 Created: 2015-10-16 Last updated: 2017-05-17Bibliographically approved
In thesis
1. Analysis of boundary and interface closures for finite difference methods for the wave equation
Open this publication in new window or tab >>Analysis of boundary and interface closures for finite difference methods for the wave equation
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

We consider high order finite difference methods for the wave equations in the second order form, where the finite difference operators satisfy the summation-by-parts principle. Boundary conditions and interface conditions are imposed weakly by the simultaneous-approximation-term method, and non-conforming grid interfaces are handled by an interface operator that is based on either interpolating directly between the grids or on projecting to piecewise continuous polynomials on an intermediate grid.

Stability and accuracy are two important aspects of a numerical method. For accuracy, we prove the convergence rate of the summation-by-parts finite difference schemes for the wave equation. Our approach is based on Laplace transforming the error equation in time, and analyzing the solution to the boundary system in the Laplace space. In contrast to first order equations, we have found that the determinant condition for the second order equation is less often satisfied for a stable numerical scheme. If the determinant condition is satisfied uniformly in the right half plane, two orders are recovered from the boundary truncation error; otherwise we perform a detailed analysis of the solution to the boundary system in the Laplace space to obtain an error estimate. Numerical experiments demonstrate that our analysis gives a sharp error estimate.

For stability, we study the numerical treatment of non-conforming grid interfaces. In particular, we have explored two interface operators: the interpolation operators and projection operators applied to the wave equation. A norm-compatible condition involving the interface operator and the norm related to the SBP operator is essential to prove stability by the energy method for first order equations. In the analysis, we have found that in contrast to first order equations, besides the norm-compatibility condition an extra condition must be imposed on the interface operators to prove stability by the energy method. Furthermore, accuracy and efficiency studies are carried out for the numerical schemes.

Place, publisher, year, edition, pages
Uppsala University, 2015
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2015-005
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-264761 (URN)
Supervisors
Available from: 2015-10-14 Created: 2015-10-16 Last updated: 2017-08-31Bibliographically approved
2. Finite Difference and Discontinuous Galerkin Methods for Wave Equations
Open this publication in new window or tab >>Finite Difference and Discontinuous Galerkin Methods for Wave Equations
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
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:nbn:se:uu:diva-320614 (URN)978-91-554-9927-3 (ISBN)
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

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Wang, SiyangKreiss, Gunilla

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