Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Well-posed and Stable Transmission Problems
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]

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 is analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to several examples including the coupling of fluid flow models, multi-grid implementations, multi-block formulations and numerical filtering.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. , 28 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 15
Keyword [en]
Initial-boundary value problems, Transmission problems, Energy estimates, Well-posedness, Multi-block, Numerical Filter. Interpolation, Multi-grid, Summation-by-Parts, Stability
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-142348ISRN: LiTH-MAT-R--2017/15--SEOAI: oai:DiVA.org:liu-142348DiVA: diva2:1153062
Available from: 2017-10-27 Created: 2017-10-27 Last updated: 2017-11-20Bibliographically 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

Open Access in DiVA

Well-posed and Stable Transmission Problems(387 kB)10 downloads
File information
File name FULLTEXT02.pdfFile size 387 kBChecksum SHA-512
38ef421f2068f8f26aa4ad1ce3c132879f32a3d0ed1856e944293f615f89d1082e8ad19e4b2dd50e35e2cb25b3fe88b2d2a718333a33369e9b7babb121d673c3
Type fulltextMimetype application/pdf

Authority records BETA

Nordström, JanLinders, Viktor

Search in DiVA

By author/editor
Nordström, JanLinders, Viktor
By organisation
Computational MathematicsFaculty of Science & Engineering
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 10 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 32 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf