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A Compact Fourth Order Time Discretization Method for the Wave Equation
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)Report (Other academic)
Place, publisher, year, edition, pages
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2003-041
National Category
Computational Mathematics
URN: urn:nbn:se:uu:diva-48120OAI: oai:DiVA.org:uu-48120DiVA: diva2:76027
Available from: 2007-02-12 Created: 2007-02-12 Last updated: 2011-11-17Bibliographically approved
In thesis
1. High Order Finite Difference Methods in Space and Time
Open this publication in new window or tab >>High Order Finite Difference Methods in Space and Time
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. 28 p.
Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1104-232X ; 880
finite difference methods, Navier-Stokes equations, high order time discretization, deferred correction, stability
National Category
Computational Mathematics
Research subject
Numerical Analysis
urn:nbn:se:uu:diva-3559 (URN)91-554-5721-5 (ISBN)
Public defence
2003-10-24, Room 1211, Polacksbacken, Uppsala University, Uppsala, 10:15 (English)
Available from: 2003-09-24 Created: 2003-09-24 Last updated: 2011-10-26Bibliographically approved

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