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Uniformly Best Wavenumber Approximations by Spatial Central Difference Operators
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
2015 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 300, 695-709 p.Article in journal (Refereed) Published
Abstract [en]

We construct accurate central difference stencils for problems involving high frequency waves or multi-frequency solutions over long time intervals with a relatively coarse spatial mesh, and with an easily obtained bound on the dispersion error. This is done by demonstrating that the problem of constructing central difference stencils that have minimal dispersion error in the infinity norm can be recast into a problem of approximating a continuous function from a finite dimensional subspace with a basis forming a Chebyshev set. In this new formulation, characterising and numerically obtaining optimised schemes can be done using established theory.

Place, publisher, year, edition, pages
Elsevier, 2015. Vol. 300, 695-709 p.
Keyword [en]
Dispersion relation; Wave propagation; Wavenumber approximation; Finite differences; Approximation theory
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-120896DOI: 10.1016/j.jcp.2015.08.005ISI: 000361573200035OAI: oai:DiVA.org:liu-120896DiVA: diva2:849400
Available from: 2015-08-28 Created: 2015-08-28 Last updated: 2017-12-04Bibliographically approved
In thesis
1. Error analysis of summation-by-parts formulations: Dispersion, transmission and accuracy
Open this publication in new window or tab >>Error analysis of summation-by-parts formulations: Dispersion, transmission and accuracy
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems.

The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times.

In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid.

In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 27 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1886
National Category
Computational Mathematics Mathematical Analysis Control Engineering Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:liu:diva-143059 (URN)10.3384/diss.diva-143059 (DOI)978-91-7685-427-3 (ISBN)
Public defence
2017-12-12, Ada Lovelace,, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2017-11-20 Created: 2017-11-20 Last updated: 2017-11-20Bibliographically approved

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