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Perfectly matched layers for second order 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.
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Numerical simulation of propagating waves in unbounded spatial domains is a challenge common to many branches of engineering and applied mathematics. Perfectly matched layers (PML) are a novel technique for simulating the absorption of waves in open domains. The equations modeling the dynamics of phenomena of interest are usually posed as differential equations (or integral equations) which must be solved at every time instant. In many application areas like general relativity, seismology and acoustics, the underlying equations are systems of second order hyperbolic partial differential equations. In numerical treatment of such problems, the equations are often rewritten as first order systems and are solved in this form. For this reason, many existing PML models have been developed for first order systems. In several studies, it has been reported that there are drawbacks with rewriting second order systems into first order systems before numerical solutions are obtained. While the theory and numerical methods for first order systems are well developed, numerical techniques to solve second order hyperbolic systems is an on-going research.

In the first part of this thesis, we construct PML equations for systems of second order hyperbolic partial differential equations in two space dimensions, focusing on the equations of linear elasto-dynamics. One advantage of this approach is that we can choose auxiliary variables such that the PML is strongly hyperbolic, thus strongly well-posed. The second is that it requires less auxiliary variables as compared to existing first order formulations. However, in continuum the stability of both first order and second order formulations are linearly equivalent. A turning point is in numerical approximations. 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 for any given resolution. The second order discretization behaves much more stably. In the second order setting instability occurs only if unstable modes are well resolved.

The second part of this thesis discusses the construction of PML equations for the time-dependent Schrödinger equation. From mathematical perspective, the Schrödinger equation is unique, in the sense that it is only first order in time but second order in space. However, with slight modifications, we carry over our ideas from the hyperbolic systems to the Schrödinger equations and derive a set of asymptotically stable PML equations. The new model can be viewed as a modified complex absorbing potential (CAP). The PML model can easily be adapted to existing codes developed for CAP by accurately discretizing the auxiliary variables and appending them accordingly. Numerical experiments are presented illustrating the accuracy and absorption properties of the new PML model.

We are hopeful that the results obtained in this thesis will find useful applications in time-dependent wave scattering calculations.

Place, publisher, year, edition, pages
Uppsala University, 2010.
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2010-004
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-124538OAI: oai:DiVA.org:uu-124538DiVA, id: diva2:317660
Supervisors
Available from: 2010-05-07 Created: 2010-05-04 Last updated: 2017-08-31Bibliographically approved
List of papers
1. Well-posed and discretely stable perfectly matched layers for elastic wave equations in second order formulation
Open this publication in new window or tab >>Well-posed and discretely stable perfectly matched layers for elastic wave equations in second order formulation
2010 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2010-004
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-117577 (URN)
Available from: 2010-02-19 Created: 2010-02-19 Last updated: 2014-06-16Bibliographically 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, p. 641-663Article 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: 2017-12-07Bibliographically approved
3. Stable perfectly matched layers for the Schrödinger equations
Open this publication in new window or tab >>Stable perfectly matched layers for the Schrödinger equations
2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, p. 287-295Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2010
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-132926 (URN)10.1007/978-3-642-11795-4_30 (DOI)000395207900030 ()978-3-642-11794-7 (ISBN)
Projects
eSSENCE
Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2018-06-16Bibliographically approved

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Duru, Kenneth

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