Numerical dispersive schemes for the nonlinear Schrödinger equation

We consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to z...

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Detalles Bibliográficos
Autores: Ignat, L.I., Zuazua, E.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2009
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/508
Acceso en línea:http://hdl.handle.net/20.500.11824/508
Access Level:acceso abierto
Palabra clave:Finite differences
Nonlinear Schrödinger equations
Strichartz estimates
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spelling Numerical dispersive schemes for the nonlinear Schrödinger equationIgnat, L.I.Zuazua, E.Finite differencesNonlinear Schrödinger equationsStrichartz estimatesWe consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with L2-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates. © 2009 Society for Industrial and Applied Mathematics.201720172009info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/508reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://www.scopus.com/inward/record.uri?eid=2-s2.0-77953685944&doi=10.1137%2f070683787&partnerID=40&md5=fa446f1a1f2a1e7c4e22df4cc439bc65Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/5082026-06-19T12:47:47Z
dc.title.none.fl_str_mv Numerical dispersive schemes for the nonlinear Schrödinger equation
title Numerical dispersive schemes for the nonlinear Schrödinger equation
spellingShingle Numerical dispersive schemes for the nonlinear Schrödinger equation
Ignat, L.I.
Finite differences
Nonlinear Schrödinger equations
Strichartz estimates
title_short Numerical dispersive schemes for the nonlinear Schrödinger equation
title_full Numerical dispersive schemes for the nonlinear Schrödinger equation
title_fullStr Numerical dispersive schemes for the nonlinear Schrödinger equation
title_full_unstemmed Numerical dispersive schemes for the nonlinear Schrödinger equation
title_sort Numerical dispersive schemes for the nonlinear Schrödinger equation
dc.creator.none.fl_str_mv Ignat, L.I.
Zuazua, E.
author Ignat, L.I.
author_facet Ignat, L.I.
Zuazua, E.
author_role author
author2 Zuazua, E.
author2_role author
dc.subject.none.fl_str_mv Finite differences
Nonlinear Schrödinger equations
Strichartz estimates
topic Finite differences
Nonlinear Schrödinger equations
Strichartz estimates
description We consider semidiscrete approximation schemes for the linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with L2-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates. © 2009 Society for Industrial and Applied Mathematics.
publishDate 2009
dc.date.none.fl_str_mv 2009
2017
2017
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dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/508
url http://hdl.handle.net/20.500.11824/508
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://www.scopus.com/inward/record.uri?eid=2-s2.0-77953685944&doi=10.1137%2f070683787&partnerID=40&md5=fa446f1a1f2a1e7c4e22df4cc439bc65
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/
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dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
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