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Absorbing Layers and Non-Reflecting Boundary Conditions for Wave Propagation Problems
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2005 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

The presence of wave motion is the defining feature in many fields of application,such as electro-magnetics, seismics, acoustics, aerodynamics,oceanography and optics. In these fields, accurate numerical simulation of wave phenomena is important for the enhanced understanding of basic phenomenon, but also in design and development of various engineering applications.

In general, numerical simulations must be confined to truncated domains, much smaller than the physical space were the wave phenomena takes place. To truncate the physical space, artificial boundaries, and corresponding boundary conditions, are introduced. There are four main classes of methods that can be used to truncate problems on unbounded or large domains: boundary integral methods, infinite element methods, non-reflecting boundary condition methods and absorbing layer methods.

In this thesis, we consider different aspects of non-reflecting boundary conditions and absorbing layers. In paper I, we construct discretely non-reflecting boundary conditions for a high order centered finite difference scheme. This is done by separating the numerical solution into spurious and physical waves, using the discrete dispersion relation.

In paper II-IV, we focus on the perfectly matched layer method, which is a particular absorbing layer method. An open issue is whether stable perfectly matched layers can be constructed for a general hyperbolic system.

In paper II, we present a stable perfectly matched layer formulation for 2 x 2 symmetric hyperbolic systems in (2 + 1) dimensions. We also show how to choose the layer parameters as functions of the coefficient matrices to guarantee stability.

In paper III, we construct a new perfectly matched layer for the simulation of elastic waves in an anisotropic media. We present theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers.

In paper IV, we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters which is applicable to all hyperbolic systems, and which we prove is well-posed and perfectly matched. We also use an automatic method, derived in paper V, for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell s equations, the linearized Euler equations, as well as arbitrary 2 x 2 systems in (2 + 1) dimensions.

In paper V, we use the method of Sturm sequences for bounding the real parts of roots of polynomials, to construct an automatic method for checking Petrowsky well-posedness of a general Cauchy problem. We prove that this method can be adapted to automatically symmetrize any well-posed problem, producing an energy estimate involving only local quantities.

Place, publisher, year, edition, pages
Stockholm: KTH , 2005. , iv, 19 p.
Series
Trita-NA, ISSN 0348-2952 ; 2005:34
Keyword [en]
PML, perfectly matched layer
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-448ISBN: 91-7178-174-9 (print)OAI: oai:DiVA.org:kth-448DiVA: diva2:12464
Public defence
2005-10-28, Salongen, KTHB, Osquars backe 31, KTH, Stockholm, 10:15
Opponent
Supervisors
Note
QC 20100830Available from: 2005-10-14 Created: 2005-10-14 Last updated: 2010-08-30Bibliographically approved
List of papers
1. Discretely nonreflecting boundary conditions for higher order centered schemes for wave equations
Open this publication in new window or tab >>Discretely nonreflecting boundary conditions for higher order centered schemes for wave equations
2003 (English)In: Proceedings of the WAVES-2003 conference, Berlin: Springer Verlag , 2003, 130-135 p.Chapter in book (Other academic)
Abstract [en]

Using the framework introduced by Rawley and Colonius [2] we construct a nonreflecting boundary condition for the one-way wave equation spatially discretized with a fourth order centered difference scheme. The boundary condition, which can be extended to arbitrary order accuracy, is shown to be well posed. Numerical simulations have been performed showing promising results.

Place, publisher, year, edition, pages
Berlin: Springer Verlag, 2003
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7433 (URN)3-540-40127-X (ISBN)
Note
QC 20100830. Konferens: : 6th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2003) JYVASKYLA, FINLAND, JUN 30-JUL 04, 2003.Available from: 2005-10-14 Created: 2005-10-14 Last updated: 2010-08-30Bibliographically approved
2. Construction of stable PMLs for general 2 x 2 symmetric hyperbolic systems
Open this publication in new window or tab >>Construction of stable PMLs for general 2 x 2 symmetric hyperbolic systems
2004 (English)In: Proceedings of the HYP2004 conference, 2004, 1-8 p.Conference paper, Published paper (Refereed)
Abstract [en]

The perfectly matched layer (PML) has emerged as animportant tool for accurately solving certain hyperbolic systems onunbounded domains. An open issue is whether stable PMLs can beconstructed in general. In this work we consider the specializationof our general PML formulation to 2 × 2 symmetric hyperbolicsystems in 2 + 1 dimensions. We show how to choose the layerparameters as functions of the coefficient matrices to guaranteestability.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7434 (URN)
Conference
The HYP2004 conference, September 13-17 2004, Osaka, Japan
Note
QC 20100830Available from: 2005-10-14 Created: 2005-10-14 Last updated: 2010-08-30Bibliographically approved
3. A New Absorbing Layer for Elastic Waves
Open this publication in new window or tab >>A New Absorbing Layer for Elastic Waves
2006 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 215, no 2, 642-660 p.Article in journal (Refereed) Published
Abstract [en]

A new perfectly matched layer (PML) for the simulation of elastic waves in anisotropic media on an unbounded domain is constructed. Theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers, are presented. In addition, the layer can be formulated with fewer auxiliary variables than the split-field PML.

Keyword
Absorbing layers, Elastic waves, Perfectly matched layers, PML
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-7435 (URN)10.1016/j.jcp.2005.11.006 (DOI)000237458200016 ()2-s2.0-33645909260 (Scopus ID)
Note
QC 20100830. Uppdaterad från Submitted till Published 20100830.Available from: 2005-10-14 Created: 2005-10-14 Last updated: 2010-08-30Bibliographically approved
4. Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability
Open this publication in new window or tab >>Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability
2006 (English)In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 67, no 1, 1-23 p.Article in journal (Refereed) Published
Abstract [en]

Since its introduction the perfectly matched layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limited to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters, which is applicable to all hyperbolic systems and which we prove is well-posed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell's equations, the linearized Euler equations, and arbitrary 2 x 2 systems in (2 + 1) dimensions.

Keyword
Perfectly matched layers, Stability, Computational methods, Euler equations, Mathematical models, Maxwell equations, Parameter estimation, Wave propagation, Computational electromagnetics, Hyperbolic systems, Mathematical analysis, Perfectly matched layers, Magnetoelectric effects
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7436 (URN)10.1137/050639107 (DOI)000243279300001 ()2-s2.0-33847744194 (Scopus ID)
Note
QC 20100830. Uppdaterad från Submitted till Published 20100830.Available from: 2005-10-14 Created: 2005-10-14 Last updated: 2010-09-02Bibliographically approved
5. Automatic symmetrization and energy estimates using local operators for partial differential equations
Open this publication in new window or tab >>Automatic symmetrization and energy estimates using local operators for partial differential equations
2007 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 32, no 7, 1129-1145 p.Article in journal (Refereed) Published
Abstract [en]

We develop a method for automatically symmetrizing Petrowsky well-posed Cauchy problems for constant coefficient linear partial differential equations. The method is rooted in the Sturm sequence technique for establishing the location of the roots of a complex polynomial and can be automated using standard symbolic computation tools. In the special case of homogeneous strictly hyperbolic scalar equations, we show that the resulting estimates are strong enough to control all principal order derivatives and thus can be used in place of the Leray energies. We also illustrate the method by applying it to various problems of mixed type.

Keyword
Cauchy problem, Energy estimates, Sturm sequences, Well-posedness
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7437 (URN)10.1080/03605300600854258 (DOI)000250012900005 ()2-s2.0-34547762609 (Scopus ID)
Note
QC 20100830. Tidigare titel: On symmetrization and energy estimates using local operators for partial differential equations. Titel ändrad samt uppdaterad från Manuskript till Artikel i tidskrift 20100830Available from: 2005-10-14 Created: 2005-10-14 Last updated: 2010-08-30Bibliographically approved

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