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Stable Numerical Methods with Boundary and Interface Treatment for Applications in 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.
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: Acta Universitatis Upsaliensis, 2012. , 26 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 985
Keyword [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:uu:diva-182953ISBN: 978-91-554-8509-2 (print)OAI: oai:DiVA.org:uu-182953DiVA: diva2:562404
Public defence
2012-12-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2012-11-16 Created: 2012-10-19 Last updated: 2013-01-23Bibliographically 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, 2659-2676 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-105956 (URN)10.1016/j.apnum.2009.06.001 (DOI)000268513700023 ()
Available from: 2009-06-10 Created: 2009-06-10 Last updated: 2013-01-23Bibliographically 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, Vol. 8, 1111-1138 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-126723 (URN)10.4208/cicp.260409.120210a (DOI)000284672100007 ()
Available from: 2010-06-23 Created: 2010-06-23 Last updated: 2013-01-23Bibliographically 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, 4216-4231 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-134055 (URN)10.1016/j.jcp.2010.11.020 (DOI)000290185000007 ()
Available from: 2010-11-20 Created: 2010-11-21 Last updated: 2013-01-23Bibliographically 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, 4867-4884 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-173565 (URN)10.1016/j.jcp.2012.04.007 (DOI)000304257600021 ()
Available from: 2012-04-18 Created: 2012-04-26 Last updated: 2013-01-23Bibliographically 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)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-032
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-183131 (URN)
Available from: 2012-10-19 Created: 2012-10-22 Last updated: 2013-01-23Bibliographically approved

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