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Boundary waves and stability of the perfectly matched layer
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.
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)Report (Other academic)
Place, publisher, year, edition, pages
2012.
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-007
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-172887OAI: oai:DiVA.org:uu-172887DiVA: diva2:515967
Available from: 2012-04-10 Created: 2012-04-16 Last updated: 2012-05-15Bibliographically approved
In thesis
1. Perfectly Matched Layers and High Order Difference Methods for Wave Equations
Open this publication in new window or tab >>Perfectly Matched Layers and High Order Difference Methods for Wave Equations
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete.

We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials.

We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML.

Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. 47 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 931
Keyword
Elastic waves, Surface waves, Perfectly matched layers, High order difference methods, Stability, Summation-by-parts operators, Boundary treatments
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-173009 (URN)978-91-554-8365-4 (ISBN)
Public defence
2012-06-08, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, VR 2009-5852
Available from: 2012-05-14 Created: 2012-04-17 Last updated: 2012-10-05Bibliographically approved

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http://www.it.uu.se/research/publications/reports/2012-007/

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Duru, KennethKreiss, Gunilla

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