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Numerical Computations with Fundamental Solutions
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. (ANLA)
2005 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Numeriska beräkningar med fundamentallösningar (Swedish)
Abstract [en]

Two solution strategies for large, sparse, and structured algebraic systems of equations are considered. The first strategy is to construct efficient preconditioners for iterative solvers. The second is to reduce the sparse algebraic system to a smaller, dense system of equations, which are called the boundary summation equations.

The proposed preconditioners perform well when applied to equations that are discretizations of linear first order partial differential equations. Analysis shows that also very simple iterative methods converge in a number of iterations that is independent of the number of unknowns, if our preconditioners are applied to certain scalar model problems. Numerical experiments indicate that this property holds also for more complicated cases, and a flow problem modeled by the nonlinear Euler equations is treated successfully.

The reduction process is applicable to a large class of difference equations. There is no approximation involved in the reduction, so the solution of the original algebraic equations is determined exactly if the reduced system is solved exactly. The reduced system is well suited for iterative solution, especially if the original system of equations is a discretization of a first order differential equation. The technique is used for several problems, ranging from scalar model problems to a semi-implicit discretization of the compressible Navier-Stokes equations.

Both strategies use the concept of fundamental solutions, either of differential or difference operators. An algorithm for computing fundamental solutions of difference operators is also presented.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2005. , p. 51
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 45
Keywords [en]
fundamental solution, partial differential equation, partial difference equation, iterative method, preconditioner, boundary method
National Category
Computational Mathematics
Research subject
Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-5757ISBN: 91-554-6227-8 (print)OAI: oai:DiVA.org:uu-5757DiVA, id: diva2:166283
Public defence
2005-05-13, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2005-04-21 Created: 2005-04-21 Last updated: 2014-09-03Bibliographically approved
List of papers
1. Preconditioners based on fundamental solutions
Open this publication in new window or tab >>Preconditioners based on fundamental solutions
2005 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 45, p. 481-494Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-78887 (URN)10.1007/s10543-005-0010-7 (DOI)
Available from: 2007-03-11 Created: 2007-03-11 Last updated: 2018-01-13Bibliographically approved
2. An algorithm for computing fundamental solutions of difference operators
Open this publication in new window or tab >>An algorithm for computing fundamental solutions of difference operators
2004 (English)In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 36, p. 331-343Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-67772 (URN)10.1007/s11075-004-2879-7 (DOI)
Available from: 2006-05-20 Created: 2006-05-20 Last updated: 2018-01-10Bibliographically approved
3. Discrete fundamental solution preconditioning for hyperbolic systems of PDE
Open this publication in new window or tab >>Discrete fundamental solution preconditioning for hyperbolic systems of PDE
2007 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 30, p. 35-60Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-84009 (URN)10.1007/s10915-005-9018-z (DOI)000243906800002 ()
Available from: 2007-01-31 Created: 2007-01-31 Last updated: 2018-01-13Bibliographically approved
4. Boundary Summation Equations
Open this publication in new window or tab >>Boundary Summation Equations
2004 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2004-042
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-68483 (URN)
Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2014-09-03Bibliographically approved
5. Navier-Stokes equations for low Mach number flows solved by boundary summation
Open this publication in new window or tab >>Navier-Stokes equations for low Mach number flows solved by boundary summation
2005 (English)Manuscript (preprint) (Other academic)
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-92953 (URN)
Available from: 2005-04-21 Created: 2005-04-21 Last updated: 2018-01-13Bibliographically approved

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