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High Order Finite Difference Methods in Space and Time
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. (Waves and Fluids)
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, high order accurate discretization schemes for partial differential equations are investigated.

In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order finite difference scheme on a staggered mesh is used. In Paper II, the analysis for the second order scheme is used to develop a fourth order scheme for the fully nonlinear Navier-Stokes equations. The fully nonlinear incompressible Navier-Stokes equations in two space dimensions are considered on an orthogonal curvilinear grid. Numerical tests are performed with a fourth order accurate Padé type spatial finite difference scheme and a semi-implicit BDF2 scheme in time.

In Papers III-V, a class of high order accurate time-discretization schemes based on the deferred correction principle is investigated. The deferred correction principle is based on iteratively eliminating lower order terms in the local truncation error, using previously calculated solutions, in each iteration obtaining more accurate solutions. It is proven that the schemes are unconditionally stable and stability estimates are given using the energy method. Error estimates and smoothness requirements are derived. Special attention is given to the implementation of the boundary conditions for PDE. The scheme is applied to a series of numerical problems, confirming the theoretical results.

In the sixth paper, a time-compact fourth order accurate time discretization for the one- and two-dimensional wave equation is considered. Unconditional stability is established and fourth order accuracy is numerically verified. The scheme is applied to a two-dimensional wave propagation problem with discontinuous coefficients.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 2003. , p. 28
Series
Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1104-232X ; 880
Keywords [en]
finite difference methods, Navier-Stokes equations, high order time discretization, deferred correction, stability
National Category
Computational Mathematics
Research subject
Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-3559ISBN: 91-554-5721-5 (print)OAI: oai:DiVA.org:uu-3559DiVA, id: diva2:163250
Public defence
2003-10-24, Room 1211, Polacksbacken, Uppsala University, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2003-09-24 Created: 2003-09-24 Last updated: 2011-10-26Bibliographically approved
List of papers
1. Boundary conditions and estimates for the linearized Navier-Stokes equations on staggered grids
Open this publication in new window or tab >>Boundary conditions and estimates for the linearized Navier-Stokes equations on staggered grids
2003 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 32, p. 1093-1112Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-41005 (URN)10.1016/S0045-7930(02)00090-7 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2017-12-06Bibliographically approved
2. A Compact Higher Order Finite Difference Method for the Incompressible Navier-Stokes Equations
Open this publication in new window or tab >>A Compact Higher Order Finite Difference Method for the Incompressible Navier-Stokes Equations
2002 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 17, p. 551-560Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-48117 (URN)10.1023/A:1015166529060 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2018-01-11Bibliographically approved
3. Deferred Correction Methods for Initial Value Problems
Open this publication in new window or tab >>Deferred Correction Methods for Initial Value Problems
2001 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 41, p. 986-995Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-40588 (URN)10.1023/A:1021937227950 (DOI)
Available from: 2007-01-25 Created: 2007-01-25 Last updated: 2018-01-11Bibliographically approved
4. Deferred Correction Methods for Initial Boundary Value Problems
Open this publication in new window or tab >>Deferred Correction Methods for Initial Boundary Value Problems
2002 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 17, p. 241-251Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-48118 (URN)10.1023/A:1015113017248 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2018-01-11Bibliographically approved
5. Error Estimates for Deferred Correction Methods in Time
Open this publication in new window or tab >>Error Estimates for Deferred Correction Methods in Time
2003 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2003-040
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-48119 (URN)
Available from: 2007-02-12 Created: 2007-02-12 Last updated: 2011-11-17Bibliographically approved
6. A Compact Fourth Order Time Discretization Method for the Wave Equation
Open this publication in new window or tab >>A Compact Fourth Order Time Discretization Method for the Wave Equation
2003 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2003-041
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-48120 (URN)
Available from: 2007-02-12 Created: 2007-02-12 Last updated: 2011-11-17Bibliographically approved

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