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Long time error bounds for the wave equation on second order form
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)Report (Other academic)
Abstract [en]

Temporal error bounds for the wave equation expressed on second order form is investigated. By using the energy method, we show that, with appropriate choices of boundary condition, the time and space derivative of the error is bounded even for long times. No long time bound on the actual error can be obtained, although numerical experiments indicate that such a bound exist.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. , p. 12
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:1
Keywords [en]
Error bounds, second order form, summation-by-parts, finite differences, simultaneous approximation terms, wave equation
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-134058ISRN: LiTH-MAT-R--2017/01--SEOAI: oai:DiVA.org:liu-134058DiVA, id: diva2:1067263
Available from: 2017-01-20 Created: 2017-01-20 Last updated: 2017-01-26Bibliographically approved
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. p. 40
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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