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Stable High-Order Finite Difference Methods for Aerodynamics
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis. (Waves and Fluids)
2004 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Stabila högordnings finita differensmetoder för aerodynamik (Swedish)
Abstract [en]

In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates.

The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version.

With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes.

We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 2004. , 25 p.
Series
Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1104-232X ; 1026
Keyword [en]
finite difference methods, high-order accuracy, summation-by-parts, stability, energy estimates, finite volume methods
National Category
Computational Mathematics
Research subject
Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-4621ISBN: 91-554-6063-1 (print)OAI: oai:DiVA.org:uu-4621DiVA: diva2:165268
Public defence
2004-11-12, Room 1211, Polacksbacken, Lägerhyddsvägen 2F, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2004-10-22 Created: 2004-10-22 Last updated: 2011-10-27Bibliographically approved
List of papers
1. On Coordinate Transformations for Summation-by-Parts Operators
Open this publication in new window or tab >>On Coordinate Transformations for Summation-by-Parts Operators
2004 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 20, 29-42 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-71164 (URN)10.1023/A:1025881528802 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-29Bibliographically approved
2. Stable and Accurate Artificial Dissipation
Open this publication in new window or tab >>Stable and Accurate Artificial Dissipation
2004 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 21, 57-79 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-71152 (URN)10.1023/B:JOMP.0000027955.75872.3f (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-29Bibliographically approved
3. Accuracy requirements for transient aerodynamics
Open this publication in new window or tab >>Accuracy requirements for transient aerodynamics
2003 (English)In: Proc. 16th AIAA CFD Conference, AIAA , 2003Conference paper (Refereed)
Place, publisher, year, edition, pages
AIAA, 2003
Series
Conference Proceeding Series, 2003-3689
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-48285 (URN)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-26Bibliographically approved
4. Steady-State Computations Using Summation-by-Parts Operators
Open this publication in new window or tab >>Steady-State Computations Using Summation-by-Parts Operators
2005 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 24, 79-95 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-78927 (URN)10.1007/s10915-004-4788-2 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-29Bibliographically approved
5. Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids
Open this publication in new window or tab >>Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids
2004 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 51, 101-125 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-71143 (URN)10.1016/j.apnum.2004.02.001 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-29Bibliographically approved
6. Artificial Dissipation for Strictly Stable Finite Volume Methods on Unstructured Meshes
Open this publication in new window or tab >>Artificial Dissipation for Strictly Stable Finite Volume Methods on Unstructured Meshes
2004 (English)In: Computational Mechanics Abstracts: Volume II, Beijing, China: Tsinghua University Press , 2004, 7-7 p.Conference paper (Other academic)
Place, publisher, year, edition, pages
Beijing, China: Tsinghua University Press, 2004
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-71104 (URN)7-302-09342-3 (ISBN)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-26Bibliographically approved
7. Well-posed boundary conditions for the Navier-Stokes equations
Open this publication in new window or tab >>Well-posed boundary conditions for the Navier-Stokes equations
2005 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 43, 1231-1255 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-78928 (URN)10.1137/040604972 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-29Bibliographically approved
8. On the Order of Accuracy for Difference Approximations of Initial-Boundary Value Problems
Open this publication in new window or tab >>On the Order of Accuracy for Difference Approximations of Initial-Boundary Value Problems
2004 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2004-040
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-71168 (URN)
Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2011-11-17Bibliographically approved

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  • apa
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