liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • 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
Error analysis of summation-by-parts formulations: Dispersion, transmission and accuracy
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 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. , p. 27
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: urn:nbn:se:liu:diva-143059DOI: 10.3384/diss.diva-143059ISBN: 978-91-7685-427-3 (print)OAI: oai:DiVA.org:liu-143059DiVA, id: diva2:1158370
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
List of papers
1. Uniformly Best Wavenumber Approximations by Spatial Central Difference Operators
Open this publication in new window or tab >>Uniformly Best Wavenumber Approximations by Spatial Central Difference Operators
2015 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 300, p. 695-709Article 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
Keyword
Dispersion relation; Wave propagation; Wavenumber approximation; Finite differences; Approximation theory
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-120896 (URN)10.1016/j.jcp.2015.08.005 (DOI)000361573200035 ()
Available from: 2015-08-28 Created: 2015-08-28 Last updated: 2017-12-04Bibliographically approved
2. A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains
Open this publication in new window or tab >>A simple and efficient incompressible Navier-Stokes solver for unsteady complex geometry flows on truncated domains
Show others...
2017 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 150, p. 84-94Article in journal (Refereed) Published
Abstract [en]

Incompressible Navier-Stokes solvers based on the projection method often require an expensive numerical solution of a Poisson equation for a pressure-like variable. This often involves linear system solvers based on iterative and multigrid methods which may limit the ability to scale to large numbers of processors. The artificial compressibility method (ACM) [6] introduces a time derivative of the pressure into the incompressible form of the continuity equation creating a coupled closed hyperbolic system that does not require a Poisson equation solution and allows for explicit time-marching and localized stencil numerical methods. Such a scheme should theoretically scale well on large numbers of CPUs, GPU'€™s, or hybrid CPU-GPU architectures. The original ACM was only valid for steady flows and dual-time stepping was often used for time-accurate simulations. Recently, Clausen [7] has proposed the entropically damped artificial compressibility (EDAC) method which is applicable to both steady and unsteady flows without the need for dual-time stepping. The EDAC scheme was successfully tested with both a finite-difference MacCormack'€™s method for the two-dimensional lid driven cavity and periodic double shear layer problem and a finite-element method for flow over a square cylinder, with scaling studies on the latter to large numbers of processors. In this study, we discretize the EDAC formulation with a new optimized high-order centered finite-difference scheme and an explicit fourth-order Runge-€“Kutta method. This is combined with an immersed boundary method to efficiently treat complex geometries and a new robust outflow boundary condition to enable higher Reynolds number simulations on truncated domains. Validation studies for the Taylor-€“Green Vortex problem and the lid driven cavity problem in both 2D and 3D are presented. An eddy viscosity subgrid-scale model is used to enable large eddy simulations for the 3D cases. Finally, an application to flow over a sphere is presented to highlight the boundary condition and performance comparisons to a traditional incompressible Navier-€“Stokes solver is shown for the 3D lid driven cavity. Overall, the combined EDAC formulation and discretization is shown to be both effective and affordable.

Place, publisher, year, edition, pages
Elsevier, 2017
Keyword
Artificial compressibility method, EDAC, High-order numerical methods, Large Eddy simulation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-136507 (URN)10.1016/j.compfluid.2017.03.030 (DOI)000401219000007 ()
Note

Funding agencies: Rosenblatt Chair within the faculty of Mechanical Engineering; Zeff Fellowship Trust

Available from: 2017-04-19 Created: 2017-04-19 Last updated: 2017-11-20Bibliographically approved
3. Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations
Open this publication in new window or tab >>Summation-by-Parts Operators with Minimal Dispersion Error for Coarse Grid Flow Calculations
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 340, p. 34p. 160-176Article in journal (Refereed) Published
Abstract [en]

We present a procedure for constructing Summation-by-Parts operators with minimal dispersion error both near and far from numerical interfaces. Examples of such operators are constructed and compared with a higher order non-optimised Summation-by-Parts operator. Experiments show that the optimised operators are superior for wave propagation and turbulent flows involving large wavenumbers, long solution times and large ranges of resolution scales.

Place, publisher, year, edition, pages
Elsevier, 2017. p. 34
Keyword
Summation-by-Parts, Dispersion relation, Finite differences, Wave Propagation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-136474 (URN)10.1016/j.jcp.2017.03.039 (DOI)000401137900009 ()
Available from: 2017-04-12 Created: 2017-04-12 Last updated: 2017-11-20Bibliographically approved
4. On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form
Open this publication in new window or tab >>On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form
2017 (English)Report (Other academic)
Abstract [en]

In this paper we generalise results regarding the order of accuracy of finite difference operators on Summation-By-Parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy 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. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. p. 15
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2017:11
Keyword
Finite difference schemes, summation-by-parts operators, numerical differentiation, quadrature rules, order of accuracy
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-140815 (URN)
Available from: 2017-09-12 Created: 2017-09-12 Last updated: 2017-11-20Bibliographically approved
5. Well-posed and Stable Transmission Problems
Open this publication in new window or tab >>Well-posed and Stable Transmission Problems
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. p. 28
Series
LiTH-MAT-R, ISSN 0348-2960 ; 15
Keyword
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:nbn:se:liu:diva-142348 (URN)LiTH-MAT-R--2017/15--SE (ISRN)
Available from: 2017-10-27 Created: 2017-10-27 Last updated: 2017-11-20Bibliographically approved

Open Access in DiVA

fulltext(824 kB)68 downloads
File information
File name FULLTEXT01.pdfFile size 824 kBChecksum SHA-512
7dcb6ebbc8961e4644a48bb59e70fd080a6ca63e08aa0882f9836a6f07e2ee5e7f74b089c077823eb8199ff958d2711d5700d546697f11cdbc0dc53e7f70e089
Type fulltextMimetype application/pdf
omslag(271 kB)6 downloads
File information
File name COVER01.pdfFile size 271 kBChecksum SHA-512
91923f039a32c6a3dd0657ae8bc13cd2abb2f06bfd4f635e02634f6d82e48c2c7276f4337f2bb07dad0f3b47de03c4a769a977a7c0db4269e47a1f95cdca7154
Type coverMimetype application/pdf

Other links

Publisher's full text

Search in DiVA

By author/editor
Linders, Viktor
By organisation
Computational MathematicsFaculty of Science & Engineering
Computational MathematicsMathematical AnalysisControl EngineeringFluid Mechanics and Acoustics

Search outside of DiVA

GoogleGoogle Scholar
Total: 68 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

doi
isbn
urn-nbn

Altmetric score

doi
isbn
urn-nbn
Total: 14498 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • 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