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Stable and High-Order Finite Difference Methods for Multiphysics Flow Problems
Uppsala universitet, Avdelningen för beräkningsvetenskap.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Stabila finita differensmetoder med hög noggrannhetsordning för multifysik- och flödesproblem (Swedish)
Abstract [en]

Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software.

We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models.

The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 2013. , 35 p.
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1004
Keyword [en]
Summation-by-parts, Simultaneous Approximation Term, Stability, High-order accuracy, Finite difference methods, Dual consistency
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
URN: urn:nbn:se:liu:diva-89130ISBN: 978-91-554-8557-3OAI: diva2:607034
Public defence
2013-02-01, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, 10:15 (English)

Avhandling framlagd vid Uppsala universitet 2013-02-01.

Available from: 2013-02-21 Created: 2013-02-21 Last updated: 2013-08-30Bibliographically approved

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Berg, Jens
Computational Mathematics

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