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A stable and conservative method for locally adapting the design order of finite difference schemes
Uppsala University.ORCID-id: 0000-0003-1216-1672
Uppsala University.
Linköping University.
2011 (Engelska)Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, nr 11, s. 4216-4231Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing Summation-By-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.

Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

Ort, förlag, år, upplaga, sidor
2011. Vol. 230, nr 11, s. 4216-4231
Nationell ämneskategori
Beräkningsmatematik Datavetenskap (datalogi)
Forskningsämne
Matematik, Tillämpad matematik
Identifikatorer
URN: urn:nbn:se:lnu:diva-77771DOI: 10.1016/j.jcp.2010.11.020ISI: 000290185000007OAI: oai:DiVA.org:lnu-77771DiVA, id: diva2:1248200
Tillgänglig från: 2018-09-14 Skapad: 2018-09-14 Senast uppdaterad: 2018-09-25Bibliografiskt granskad
Ingår i avhandling
1. Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
Öppna denna publikation i ny flik eller fönster >>Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala: Uppsala University, 2012. s. 26
Serie
Acta Universitatis Upsaliensis : Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 985
Nyckelord
summation by parts, simultaneous approximation term, accuracy, stability, finite difference methods
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
urn:nbn:se:lnu:diva-77769 (URN)978-91-554-8509-2 (ISBN)
Disputation
2012-12-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, 10:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2018-09-18 Skapad: 2018-09-14 Senast uppdaterad: 2018-10-23Bibliografiskt granskad

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