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A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-7972-6183
Department of Information Technology, Division of Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden.
Department of Geological Sciences, San Diego State University, San Diego, CA 92182-1020, USA.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
2014 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 16, no 2, 541-570 p.Article in journal (Refereed) Published
Abstract [en]

A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.

Place, publisher, year, edition, pages
Global Science Press, 2014. Vol. 16, no 2, 541-570 p.
Keyword [en]
Summation-by-parts, weak boundary conditions, penalty technique, high-order accuracy, finite difference schemes, stability, steady-state, non-reflecting boundary conditions.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-107508DOI: 10.4208/cicp.020313.120314aISI: 000340786500011OAI: oai:DiVA.org:liu-107508DiVA: diva2:729882
Available from: 2014-06-26 Created: 2014-06-13 Last updated: 2017-12-05
In thesis
1. High-order finite difference approximations for hyperbolic problems: multiple penalties and non-reflecting boundary conditions
Open this publication in new window or tab >>High-order finite difference approximations for hyperbolic problems: multiple penalties and non-reflecting boundary conditions
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations.

For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT).

A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time.

To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time.

In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 40 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1824
National Category
Computational Mathematics Control Engineering Signal Processing Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:liu:diva-134127 (URN)10.3384/diss.diva-134127 (DOI)9789176855959 (ISBN)
Public defence
2017-02-21, Visionen, Hus B, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2017-01-24 Created: 2017-01-24 Last updated: 2017-09-13Bibliographically approved

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A flexible boundary procedure for hyperbolic problems: multiple penalty terms applied in a domain.(1127 kB)147 downloads
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Nordström, JanFrenander, Hannes

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