Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce a...

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Detalhes bibliográficos
Autores: Marica, A., Zuazua, E.
Formato: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2014
País:España
Recursos:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/237
Acesso em linha:http://hdl.handle.net/20.500.11824/237
Access Level:acceso abierto
Palavra-chave:Hamiltonians
Inverse problems
Mathematical transformations
Analysis and simulation
Diffeomorphic transformation
Finite difference approximations
Finite difference scheme
Non-uniform mesh
Propagation properties
Variable coefficients
Wigner transforms
Finite difference method
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spelling Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure ApproachMarica, A.Zuazua, E.HamiltoniansInverse problemsMathematical transformationsAnalysis and simulationDiffeomorphic transformationFinite difference approximationsFinite difference schemeNon-uniform meshPropagation propertiesVariable coefficientsWigner transformsFinite difference methodIn this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semiclassical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for the class (Formula presented.) of globally Lipschitz coefficients and non-uniform grids obtained by means of (Formula presented.)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space‚Äìtime projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory.201620162014info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/237reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttp://link.springer.com/article/10.1007%2Fs10208-014-9232-xinfo:eu-repo/grantAgreement/EC/FP7/246775info:eu-repo/grantAgreement/MICINN//MTM2011-29306-C02-01Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/2372026-06-19T12:47:47Z
dc.title.none.fl_str_mv Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
title Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
spellingShingle Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
Marica, A.
Hamiltonians
Inverse problems
Mathematical transformations
Analysis and simulation
Diffeomorphic transformation
Finite difference approximations
Finite difference scheme
Non-uniform mesh
Propagation properties
Variable coefficients
Wigner transforms
Finite difference method
title_short Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
title_full Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
title_fullStr Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
title_full_unstemmed Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
title_sort Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach
dc.creator.none.fl_str_mv Marica, A.
Zuazua, E.
author Marica, A.
author_facet Marica, A.
Zuazua, E.
author_role author
author2 Zuazua, E.
author2_role author
dc.subject.none.fl_str_mv Hamiltonians
Inverse problems
Mathematical transformations
Analysis and simulation
Diffeomorphic transformation
Finite difference approximations
Finite difference scheme
Non-uniform mesh
Propagation properties
Variable coefficients
Wigner transforms
Finite difference method
topic Hamiltonians
Inverse problems
Mathematical transformations
Analysis and simulation
Diffeomorphic transformation
Finite difference approximations
Finite difference scheme
Non-uniform mesh
Propagation properties
Variable coefficients
Wigner transforms
Finite difference method
description In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semiclassical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for the class (Formula presented.) of globally Lipschitz coefficients and non-uniform grids obtained by means of (Formula presented.)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space–time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory.
publishDate 2014
dc.date.none.fl_str_mv 2014
2016
2016
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/237
url http://hdl.handle.net/20.500.11824/237
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv http://link.springer.com/article/10.1007%2Fs10208-014-9232-x
info:eu-repo/grantAgreement/EC/FP7/246775
info:eu-repo/grantAgreement/MICINN//MTM2011-29306-C02-01
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
repository.name.fl_str_mv
repository.mail.fl_str_mv
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