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Summation-by-Parts Operators for High Order Finite Difference Methods
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)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 [en]
finite difference methods, accuracy, stability, dissipation
National Category
Computational Mathematics
Research subject
Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-3434ISBN: 91-554-5596-4 (print)OAI: oai:DiVA.org:uu-3434DiVA: diva2:162794
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
List of papers
1. Boundary Procedures for Summation-by-Parts Operators
Open this publication in new window or tab >>Boundary Procedures for Summation-by-Parts Operators
2003 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 18, 133-153 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-80568 (URN)10.1023/A:1020342429644 (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, Published 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. Finite Difference Approximations of Second Derivatives on Summation by Parts Form
Open this publication in new window or tab >>Finite Difference Approximations of Second Derivatives on Summation by Parts Form
2003 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2003-012
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-49027 (URN)
Available from: 2007-02-03 Created: 2007-02-03 Last updated: 2011-11-17Bibliographically approved
5. Steady State Computations Using Summation-by-Parts Operators
Open this publication in new window or tab >>Steady State Computations Using Summation-by-Parts Operators
2003 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2003-018
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-49039 (URN)
Available from: 2007-02-05 Created: 2007-02-05 Last updated: 2011-11-17Bibliographically approved

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