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High-order finite difference approximations for hyperbolicproblems: multiple penalties and non-reflecting boundary conditions
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
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: urn:nbn:se:liu:diva-134127DOI: 10.3384/diss.diva-134127ISBN: 9789176855959 (print)OAI: oai:DiVA.org:liu-134127DiVA: diva2:1068069
Public defence
2017-02-21, Visionen, Hus B, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2017-01-24 Created: 2017-01-24Bibliographically approved
List of papers
1. A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain
Open this publication in new window or tab >>A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain
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
Keyword
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:nbn:se:liu:diva-107508 (URN)10.4208/cicp.020313.120314a (DOI)000340786500011 ()
Available from: 2014-06-26 Created: 2014-06-13 Last updated: 2017-01-24
2. A Stable and Accurate Davies-like Relaxation Procedure using Multiple Penalty Terms for Lateral Boundary Conditions
Open this publication in new window or tab >>A Stable and Accurate Davies-like Relaxation Procedure using Multiple Penalty Terms for Lateral Boundary Conditions
2016 (English)In: Dynamics of atmospheres and oceans (Print), ISSN 0377-0265, E-ISSN 1872-6879, Vol. 73, 34-46 p.Article in journal (Refereed) Published
Abstract [en]

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain.

Place, publisher, year, edition, pages
Elsevier, 2016
Keyword
Boundary conditions, Boundary layers, Summation-by-parts, Energy method, Convergence, Shallow water equations
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-123117 (URN)10.1016/j.dynatmoce.2015.11.002 (DOI)000374368600003 ()
Note

Funding agencies: Swedish e-science Research Center (SeRC)

Available from: 2015-12-04 Created: 2015-12-04 Last updated: 2017-01-24Bibliographically approved
3. A stable and accurate data assmimilation technique using multiple penalty terms in space and time
Open this publication in new window or tab >>A stable and accurate data assmimilation technique using multiple penalty terms in space and time
2016 (English)Report (Other academic)
Abstract [en]

A new method for data assimilation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. 15 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 18
Keyword
data assimilation, summation-by-parts, weak boundary conditions, multiple penalties, stability, finite differences
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-133363 (URN)LiTH-MAT-R--2016/18--SE (ISRN)
Available from: 2016-12-22 Created: 2016-12-22 Last updated: 2017-01-24Bibliographically approved
4. Constructing non-reflecting boundary conditions using summation-by-parts in time
Open this publication in new window or tab >>Constructing non-reflecting boundary conditions using summation-by-parts in time
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 331, 38-48 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments.

Place, publisher, year, edition, pages
Elsevier, 2017
Keyword
Non-reflecting boundary conditions, Summation-by-parts, Simultaneous approximation terms, Finite differences, Stability, Accuracy
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-133853 (URN)10.1016/j.jcp.2016.11.038 (DOI)000393250700003 ()
Available from: 2017-01-12 Created: 2017-01-12 Last updated: 2017-05-03Bibliographically approved
5. Long time error bounds for the wave equation on second order form
Open this publication in new window or tab >>Long time error bounds for the wave equation on second order form
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. 12 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:1
Keyword
Error bounds, second order form, summation-by-parts, finite differences, simultaneous approximation terms, wave equation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-134058 (URN)LiTH-MAT-R--2017/01--SE (ISRN)
Available from: 2017-01-20 Created: 2017-01-20 Last updated: 2017-01-26Bibliographically approved

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