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On the Order of Accuracy for Difference Approximations of Initial-Boundary Value Problems
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)
2004 (English)Report (Other academic)
Place, publisher, year, edition, pages
2004.
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2004-040
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-71168OAI: oai:DiVA.org:uu-71168DiVA: diva2:99079
Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2011-11-17Bibliographically approved
In thesis
1. 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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Nordström, Jan

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