Numerical approximations for a nonlocal evolution equation

In this paper we study numerical approximations of continuous solutions to the nonlocal p-Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)|u(t, y) − u(t, x)| p−2(u(t, y) − u(t, x)) dy. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutio...

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Authors: Pérez Llanos, Mayte, Rossi, Julio D.
Format: article
Status:Published version
Publication Date:2011
Country:España
Institution:Universidad de Sevilla (US)
Repository:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/142224
Online Access:https://hdl.handle.net/11441/142224
https://doi.org/10.1137/110823559
Access Level:Open access
Keyword:numerical approximations
nonlocal diffusion
p-Laplacian
Neumann boundary conditions
sandpiles
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oai_identifier_str oai:idus.us.es:11441/142224
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repository_id_str
spelling Numerical approximations for a nonlocal evolution equationPérez Llanos, MayteRossi, Julio D.numerical approximationsnonlocal diffusionp-LaplacianNeumann boundary conditionssandpilesIn this paper we study numerical approximations of continuous solutions to the nonlocal p-Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)|u(t, y) − u(t, x)| p−2(u(t, y) − u(t, x)) dy. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties of the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity. Next, we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties. In addition, we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile. Finally, we present some numerical experiments that illustrate our results.SIAMEcuaciones Diferenciales y Análisis NuméricoFQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software2011info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/142224https://doi.org/10.1137/110823559reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésSIAM Journal on numerical analysis, 49 (5/6), 2103-2123.https://doi.org/10.1137/110823559info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1422242026-06-17T12:51:07Z
dc.title.none.fl_str_mv Numerical approximations for a nonlocal evolution equation
title Numerical approximations for a nonlocal evolution equation
spellingShingle Numerical approximations for a nonlocal evolution equation
Pérez Llanos, Mayte
numerical approximations
nonlocal diffusion
p-Laplacian
Neumann boundary conditions
sandpiles
title_short Numerical approximations for a nonlocal evolution equation
title_full Numerical approximations for a nonlocal evolution equation
title_fullStr Numerical approximations for a nonlocal evolution equation
title_full_unstemmed Numerical approximations for a nonlocal evolution equation
title_sort Numerical approximations for a nonlocal evolution equation
dc.creator.none.fl_str_mv Pérez Llanos, Mayte
Rossi, Julio D.
author Pérez Llanos, Mayte
author_facet Pérez Llanos, Mayte
Rossi, Julio D.
author_role author
author2 Rossi, Julio D.
author2_role author
dc.contributor.none.fl_str_mv Ecuaciones Diferenciales y Análisis Numérico
FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
dc.subject.none.fl_str_mv numerical approximations
nonlocal diffusion
p-Laplacian
Neumann boundary conditions
sandpiles
topic numerical approximations
nonlocal diffusion
p-Laplacian
Neumann boundary conditions
sandpiles
description In this paper we study numerical approximations of continuous solutions to the nonlocal p-Laplacian type diffusion equation, ut(t, x) = Ω J(x − y)|u(t, y) − u(t, x)| p−2(u(t, y) − u(t, x)) dy. First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties of the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition as t goes to infinity. Next, we also discretize the time variable and present a totally discrete method which also enjoys the above mentioned properties. In addition, we investigate the limit as p goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile. Finally, we present some numerical experiments that illustrate our results.
publishDate 2011
dc.date.none.fl_str_mv 2011
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/142224
https://doi.org/10.1137/110823559
url https://hdl.handle.net/11441/142224
https://doi.org/10.1137/110823559
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv SIAM Journal on numerical analysis, 49 (5/6), 2103-2123.
https://doi.org/10.1137/110823559
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv SIAM
publisher.none.fl_str_mv SIAM
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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