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On some Numerical Methods and Solution Techniques for Incompressible Flow Problems
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
2012 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Uppsala: Acta Universitatis Upsaliensis, 2012. , s. 56
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 954
HSV kategori
Forskningsprogram
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
URN: urn:nbn:se:uu:diva-179410ISBN: 978-91-554-8429-3 (tryckt)OAI: oai:DiVA.org:uu-179410DiVA, id: diva2:544511
Disputas
2012-09-24, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 14:00 (engelsk)
Opponent
Veileder
Tilgjengelig fra: 2012-09-03 Laget: 2012-08-14 Sist oppdatert: 2013-01-22bibliografisk kontrollert
Delarbeid
1. Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
Åpne denne publikasjonen i ny fane eller vindu >>Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
2010 (engelsk)Inngår i: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2010, s. 108-115Konferansepaper, Publicerat paper (Fagfellevurdert)
sted, utgiver, år, opplag, sider
Berlin: Springer-Verlag, 2010
Serie
Lecture Notes in Computer Science ; 5910
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-125577 (URN)10.1007/978-3-642-12535-5_11 (DOI)000278091900011 ()978-3-642-12534-8 (ISBN)
Tilgjengelig fra: 2010-05-10 Laget: 2010-05-24 Sist oppdatert: 2018-01-12bibliografisk kontrollert
2. On an augmented Lagrangian-based preconditioning of Oseen type problems
Åpne denne publikasjonen i ny fane eller vindu >>On an augmented Lagrangian-based preconditioning of Oseen type problems
2011 (engelsk)Inngår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, s. 865-888Artikkel i tidsskrift (Fagfellevurdert) Published
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-156201 (URN)10.1007/s10543-011-0334-4 (DOI)000297362000005 ()
Tilgjengelig fra: 2011-06-07 Laget: 2011-07-17 Sist oppdatert: 2019-01-22bibliografisk kontrollert
3. Preconditioning the incompressible Navier-Stokes equations with variable viscosity
Åpne denne publikasjonen i ny fane eller vindu >>Preconditioning the incompressible Navier-Stokes equations with variable viscosity
2012 (engelsk)Inngår i: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 30, s. 461-482Artikkel i tidsskrift (Fagfellevurdert) Published
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-179408 (URN)10.4208/jcm.1201-m3848 (DOI)000311467400002 ()
Tilgjengelig fra: 2012-09-24 Laget: 2012-08-14 Sist oppdatert: 2017-12-07bibliografisk kontrollert
4. Efficiently parallel implementation of the inverse Sherman–Morrison algorithm
Åpne denne publikasjonen i ny fane eller vindu >>Efficiently parallel implementation of the inverse Sherman–Morrison algorithm
2012 (engelsk)Rapport (Annet vitenskapelig)
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-017
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-179219 (URN)
Prosjekter
UPMARCeSSENCE
Tilgjengelig fra: 2012-08-08 Laget: 2012-08-09 Sist oppdatert: 2018-01-12bibliografisk kontrollert
5. On preconditioning incompressible non-Newtonian flow problems
Åpne denne publikasjonen i ny fane eller vindu >>On preconditioning incompressible non-Newtonian flow problems
2012 (engelsk)Rapport (Annet vitenskapelig)
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-016
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-179218 (URN)
Tilgjengelig fra: 2012-08-07 Laget: 2012-08-09 Sist oppdatert: 2018-01-12bibliografisk kontrollert
6. Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity
Åpne denne publikasjonen i ny fane eller vindu >>Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity
2012 (engelsk)Rapport (Annet vitenskapelig)
Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-019
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-179351 (URN)
Tilgjengelig fra: 2012-08-12 Laget: 2012-08-13 Sist oppdatert: 2012-09-12bibliografisk kontrollert

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