uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Perfectly Matched Layers and High Order Difference Methods for Wave Equations
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)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 [en]
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: urn:nbn:se:uu:diva-173009ISBN: 978-91-554-8365-4 OAI: oai:DiVA.org:uu-173009DiVA: diva2:516613
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
List of papers
1. A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation
Open this publication in new window or tab >>A well-posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation
2012 (English)In: Communications in Computational Physics, ISSN 1815-2406, Vol. 11, 1643-1672 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-167019 (URN)10.4208/cicp.120210.240511a (DOI)000303761800009 ()
Available from: 2012-01-12 Created: 2012-01-19 Last updated: 2012-05-31Bibliographically approved
2. Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation
Open this publication in new window or tab >>Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation
2013 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, 641-663 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-172998 (URN)10.1007/s10543-013-0426-4 (DOI)000323729800005 ()
Available from: 2013-03-19 Created: 2012-04-17 Last updated: 2014-06-16Bibliographically approved
3. On the accuracy and stability of the perfectly matched layer in transient waveguides
Open this publication in new window or tab >>On the accuracy and stability of the perfectly matched layer in transient waveguides
2012 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 53, 642-671 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-172993 (URN)10.1007/s10915-012-9594-7 (DOI)000311400300008 ()
Available from: 2012-04-23 Created: 2012-04-17 Last updated: 2013-01-02Bibliographically approved
4. Boundary waves and stability of the perfectly matched layer
Open this publication in new window or tab >>Boundary waves and stability of the perfectly matched layer
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-007
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-172887 (URN)
Available from: 2012-04-10 Created: 2012-04-16 Last updated: 2012-05-15Bibliographically approved
5. Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides
Open this publication in new window or tab >>Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-008
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-172888 (URN)
Available from: 2012-04-11 Created: 2012-04-16 Last updated: 2012-05-15Bibliographically approved
6. Stable and high-order accurate boundary treatments for the elastic wave equation on second-order form
Open this publication in new window or tab >>Stable and high-order accurate boundary treatments for the elastic wave equation on second-order form
2014 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, A2787-A2818 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-173003 (URN)10.1137/130947210 (DOI)000346838800013 ()
Available from: 2014-12-10 Created: 2012-04-17 Last updated: 2015-01-27Bibliographically approved

Open Access in DiVA

fulltext(3039 kB)1254 downloads
File information
File name FULLTEXT01.pdfFile size 3039 kBChecksum SHA-512
2b697eadb97517bd761dfe65d9f3a79a504f667c14ddae2d01e49ff505b862d137c5bfdb5aa0a8d312cafc924cefefb9807ec8723bda106b26b353305cad9488
Type fulltextMimetype application/pdf
Buy this publication >>

Search in DiVA

By author/editor
Duru, Kenneth
By organisation
Division of Scientific ComputingNumerical Analysis
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 1254 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 1355 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf