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On some Numerical Methods and Solution Techniques for Incompressible Flow 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.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The focus of this work is on numerical solution methods for solving the incompressible Navier-Stokes equations, which consist of a set of coupled nonlinear partial differential equations.

In general, after linearization and finite element discretization in space, the original nonlinear problem is converted into finding the solutions of a sequence of linear systems of equations. Because of the underlying mathematical model, the coefficient matrix of the linear system is indefinite and nonsymmetric of two-by-two block structure. Due to their less demands for computer resources than direct methods, iterative solution methods are chosen to solve these linear systems. In order to accelerate the convergence rate of the iterative methods, efficient preconditioning techniques become essential. How to construct numerically efficient preconditioners for two-by-two block systems arising in the incompressible Navier-Stokes equations has been studied intensively during the past decades, and is also a main concern in this thesis.

The Navier-Stokes equations depend on various problem parameters, such as density and viscosity, that themselves may vary in time and space as in multiphase systems. In this thesis we follow the following strategy. First, we consider the stationary Navier-Stokes equations with constant viscosity and density, and contribute to the search of efficient preconditioners by analyzing and testing the element-by-element approximation method of the Schur complement matrix and the so-called augmented Lagrangian method. Second, the variation of the viscosity is an important factor and affects the behavior of the already known preconditioners, proposed for two-by-two block matrices. To this end, we choose the augmented Lagrangian method and analyse the impact of the variation of the viscosity on the resulting preconditioner. Finally, we consider the Navier-Stokes equations with their full complexity, namely, time dependence, variable density and variable viscosity. Fast and reliable solution methods are constructed based on a reformulation of the original equations and some operator splitting techniques. Preconditioners for the so-arising linear systemsare also analyzed and tested.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. , 56 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 954
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-179410ISBN: 978-91-554-8429-3 (print)OAI: oai:DiVA.org:uu-179410DiVA: diva2:544511
Public defence
2012-09-24, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 14:00 (English)
Opponent
Supervisors
Available from: 2012-09-03 Created: 2012-08-14 Last updated: 2013-01-22Bibliographically approved
List of papers
1. Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
Open this publication in new window or tab >>Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
2010 (English)In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2010, 108-115 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2010
Series
Lecture Notes in Computer Science, 5910
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-125577 (URN)10.1007/978-3-642-12535-5_11 (DOI)000278091900011 ()978-3-642-12534-8 (ISBN)
Available from: 2010-05-10 Created: 2010-05-24 Last updated: 2012-09-12Bibliographically approved
2. On an augmented Lagrangian-based preconditioning of Oseen type problems
Open this publication in new window or tab >>On an augmented Lagrangian-based preconditioning of Oseen type problems
2011 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, 865-888 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-156201 (URN)10.1007/s10543-011-0334-4 (DOI)000297362000005 ()
Available from: 2011-06-07 Created: 2011-07-17 Last updated: 2017-12-08Bibliographically approved
3. Preconditioning the incompressible Navier-Stokes equations with variable viscosity
Open this publication in new window or tab >>Preconditioning the incompressible Navier-Stokes equations with variable viscosity
2012 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 30, 461-482 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-179408 (URN)10.4208/jcm.1201-m3848 (DOI)000311467400002 ()
Available from: 2012-09-24 Created: 2012-08-14 Last updated: 2017-12-07Bibliographically approved
4. Efficiently parallel implementation of the inverse Sherman–Morrison algorithm
Open this publication in new window or tab >>Efficiently parallel implementation of the inverse Sherman–Morrison algorithm
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-017
National Category
Computer Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-179219 (URN)
Projects
UPMARCeSSENCE
Available from: 2012-08-08 Created: 2012-08-09 Last updated: 2013-11-29Bibliographically approved
5. On preconditioning incompressible non-Newtonian flow problems
Open this publication in new window or tab >>On preconditioning incompressible non-Newtonian flow problems
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-016
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-179218 (URN)
Available from: 2012-08-07 Created: 2012-08-09 Last updated: 2012-09-12Bibliographically approved
6. Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity
Open this publication in new window or tab >>Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-019
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-179351 (URN)
Available from: 2012-08-12 Created: 2012-08-13 Last updated: 2012-09-12Bibliographically approved

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