lnu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • 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
Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
Uppsala University.ORCID iD: 0000-0003-1216-1672
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In numerical simulations, problems stemming from aerodynamics pose many challenges for the method used. Some of these are addressed in this thesis, such as the fluid interacting with objects, the presence of shocks, and various types of boundary conditions.

Scenarios of the kind mentioned above are described mathematically by initial boundary value problems (IBVPs). We discretize the IBVPs using high order accurate finite difference schemes on summation by parts form (SBP), combined with weakly imposed boundary conditions, a technique called simultaneous approximation term (SAT). By using the energy method, stability can be shown.

The weak implementation is compared to the more commonly used strong implementation, and it is shown that the weak technique enhances the rate of convergence to steady state for problems with solid wall boundary conditions. The analysis is carried out for a linear problem and supported numerically by simulations of the fully non-linear Navier–Stokes equations.

Another aspect of the boundary treatment is observed for fluid structure interaction problems. When exposed to eigenfrequencies, the coupled system starts oscillating, a phenomenon called flutter. We show that the strong implementation sometimes cause instabilities that can be mistaken for flutter.

Most numerical schemes dealing with flows including shocks are first order accurate to avoid spurious oscillations in the solution. By modifying the SBP-SAT technique, a conservative and energy stable scheme is derived where the order of accuracy can be lowered locally. The new scheme is coupled to a shock-capturing scheme and it retains the high accuracy in smooth regions.

For problems with complicated geometry, one strategy is to couple the finite difference method to the finite volume method. We analyze the accuracy of the latter on unstructured grids. For grids of bad quality the truncation error can be of zeroth order, indicating that the method is inconsistent, but we show that some of the accuracy is recovered.

We also consider artificial boundary closures on unbounded domains. Non-reflecting boundary conditions for an incompletely parabolic problem are derived, and it is shown that they yield well-posedness. The SBP-SAT methodology is employed, and we prove that the discretized problem is stable.

Place, publisher, year, edition, pages
Uppsala: Uppsala University, 2012. , p. 26
Series
Acta Universitatis Upsaliensis : Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 985
Keywords [en]
summation by parts, simultaneous approximation term, accuracy, stability, finite difference methods
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:lnu:diva-77769ISBN: 978-91-554-8509-2 (print)OAI: oai:DiVA.org:lnu-77769DiVA, id: diva2:1248194
Public defence
2012-12-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, 10:15 (English)
Opponent
Supervisors
Available from: 2018-09-18 Created: 2018-09-14 Last updated: 2018-10-23Bibliographically approved
List of papers
1. Analysis of the order of accuracy for node-centered finite volume schemes
Open this publication in new window or tab >>Analysis of the order of accuracy for node-centered finite volume schemes
2009 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 59, no 10, p. 2659-2676Article in journal (Refereed) Published
Abstract [en]

The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a elliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyzed and verified numerically. We show that the error contribution from the primal and dual grid can be treated separately.

National Category
Computational Mathematics Computer Sciences
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77773 (URN)10.1016/j.apnum.2009.06.001 (DOI)000268513700023 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-09-25Bibliographically approved
2. Fluid structure interaction problems: the necessity of a well posed, stable and accurate formulation
Open this publication in new window or tab >>Fluid structure interaction problems: the necessity of a well posed, stable and accurate formulation
2010 (English)In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 8, no 5, p. 1111-1138Article in journal (Refereed) Published
Abstract [en]

We investigate problems of fluid structure interaction type and aim for a formulation that leads to a well posed problem and a stable numerical procedure. Our first objective is to investigate if the generally accepted formulations of the fluid structure interaction problem are the only possible ones. Our second objective is to derive a stable numerical coupling. To accomplish that we will use a weak coupling procedure and employ summation-by-parts operators and penalty terms. We compare the weak coupling with other common procedures. We also study the effect of high order accurate schemes. In multiple dimensions this is a formidable task and we start by investigating the simplest possible model problem available. As a flow model we use the linearized Euler equations in one dimension and as the structure model we consider a spring.

National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77772 (URN)10.4208/cicp.260409.120210a (DOI)000284672100007 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-03Bibliographically approved
3. A stable and conservative method for locally adapting the design order of finite difference schemes
Open this publication in new window or tab >>A stable and conservative method for locally adapting the design order of finite difference schemes
2011 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 11, p. 4216-4231Article in journal (Refereed) Published
Abstract [en]

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

National Category
Computational Mathematics Computer Sciences
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77771 (URN)10.1016/j.jcp.2010.11.020 (DOI)000290185000007 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-09-25Bibliographically approved
4. Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations
Open this publication in new window or tab >>Weak and strong wall boundary procedures and convergence to steady-state of the Navier-Stokes equations
2012 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 231, no 14, p. 4867-4884Article in journal (Refereed) Published
Abstract [en]

We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier–Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77770 (URN)10.1016/j.jcp.2012.04.007 (DOI)000304257600021 ()
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-10-03Bibliographically approved
5. Exact non-reflecting boundary conditions revisited: well-posedness and stability
Open this publication in new window or tab >>Exact non-reflecting boundary conditions revisited: well-posedness and stability
2012 (English)Report (Other academic)
Abstract [en]

Exact non-reflecting boundary conditions for an incompletely parabolic system have been studied. It is shown that well-posedness is a fundamental property of the non-reflecting boundary conditions. By using summation by parts operators for the numerical approximation and a weak boundary implementation, energy stability follows automatically. The stability in combination with the high order accuracy results in a reliable, efficient and accurate method. The theory is supported by numerical simulations.

Place, publisher, year, edition, pages
Uppsala University, 2012. p. 31
Series
Technical Report ; 2012-032
National Category
Computational Mathematics
Research subject
Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-77780 (URN)
Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2019-08-30Bibliographically approved

Open Access in DiVA

Fulltext(887 kB)7 downloads
File information
File name FULLTEXT01.pdfFile size 887 kBChecksum SHA-512
629f2dcd6f731d41ada3f91f382c37cf8f2a3ac18ef1d80eca19aab2ae55b59728f4f007445835cd58c60599a48a3eff9a1b1d88fedba795be03e26c0a54e9e7
Type fulltextMimetype application/pdf

Other links

Fulltext

Authority records BETA

Eriksson, Sofia

Search in DiVA

By author/editor
Eriksson, Sofia
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 7 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

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 13 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • 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