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Automatic symmetrization and energy estimates using local operators for partial differential equations
Department of Mathematics and Statistics, University of New Mexico.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
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.

Place, publisher, year, edition, pages
2007. Vol. 32, no 7, 1129-1145 p.
Keyword [en]
Cauchy problem, Energy estimates, Sturm sequences, Well-posedness
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7437DOI: 10.1080/03605300600854258ISI: 000250012900005Scopus ID: 2-s2.0-34547762609OAI: oai:DiVA.org:kth-7437DiVA: diva2:12463
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: 2017-12-14Bibliographically approved
In thesis
1. Absorbing Layers and Non-Reflecting Boundary Conditions for Wave Propagation Problems
Open this publication in new window or tab >>Absorbing Layers and Non-Reflecting Boundary Conditions for Wave Propagation Problems
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
PML, perfectly matched layer
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-448 (URN)91-7178-174-9 (ISBN)
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

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