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Accuracy requirements for transient 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)
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)
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)
2003 (English)In: Proc. 16th AIAA CFD Conference, AIAA , 2003Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
AIAA , 2003.
Series
Conference Proceeding Series, 2003-3689
National Category
Computational Mathematics Computer Science
Identifiers
URN: urn:nbn:se:uu:diva-48285OAI: oai:DiVA.org:uu-48285DiVA: diva2:76191
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2011-11-26Bibliographically approved
In thesis
1. Summation-by-Parts Operators for High Order Finite Difference Methods
Open this publication in new window or tab >>Summation-by-Parts Operators for High Order Finite Difference Methods
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

High order accurate finite difference methods for hyperbolic and parabolic initial boundary value problems (IBVPs) are considered. Particular focus is on time dependent wave propagating problems in complex domains. Typical applications are acoustic and electromagnetic wave propagation and fluid dynamics. To solve such problems efficiently a strictly stable, high order accurate method is required.

Our recipe to obtain such schemes is to: i) Approximate the (first and second) derivatives of the IBVPs with central finite difference operators, that satisfy a summation by parts (SBP) formula. ii) Use specific procedures for implementation of boundary conditions, that preserve the SBP property. iii) Add artificial dissipation. iv) Employ a multi block structure.

Stable schemes for weakly nonlinear IBVPs require artificial dissipation to absorb the energy of the unresolved modes. This led to the construction of accurate and efficient artificial dissipation operators of SBP type, that preserve the energy and error estimate of the original problem.

To solve problems on complex geometries, the computational domain is broken up into a number of smooth and structured meshes, in a multi block fashion. A stable and high order accurate approximation is obtained by discretizing each subdomain using SBP operators and using the Simultaneous Approximation Term (SAT) procedure for both the (external) boundary and the (internal) interface conditions.

Steady and transient aerodynamic calculations around an airfoil were performed, where the first derivative SBP operators and the new artificial dissipation operators were combined to construct high order accurate upwind schemes. The computations showed that for time dependent problems and fine structures, high order methods are necessary to accurately compute the solution, on reasonably fine grids.

The construction of high order accurate SBP operators for the second derivative is one of the considerations in this thesis. It was shown that the second derivative operators could be closed with two order less accuracy at the boundaries and still yield design order of accuracy, if an energy estimate could be obtained.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2003. 23 p.
Series
Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1104-232X ; 828
Keyword
finite difference methods, accuracy, stability, dissipation
National Category
Computational Mathematics
Research subject
Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-3434 (URN)91-554-5596-4 (ISBN)
Public defence
2003-05-09, Room 2146, Polacksbacken, Uppsala University, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2003-04-17 Created: 2003-04-17 Last updated: 2011-10-27Bibliographically approved
2. Stable High-Order Finite Difference Methods for Aerodynamics
Open this publication in new window or tab >>Stable High-Order Finite Difference Methods for Aerodynamics
2004 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Stabila högordnings finita differensmetoder för aerodynamik
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
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:nbn:se:uu:diva-4621 (URN)91-554-6063-1 (ISBN)
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

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Mattsson, KenNordström, Jan

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