A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the emb...

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Autores: Petras, A., Ling, L., Piret, C., Ruuth, S.J.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2018
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/909
Acceso en línea:http://hdl.handle.net/20.500.11824/909
Access Level:acceso abierto
Palabra clave:partial differential equations on moving surfaces
closest point method
grid based particle method
radial basis functions finite differences (RBF-FD)
least-squares method
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spelling A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces Petras, A. Ling, L. Piret, C. Ruuth, S.J. partial differential equations on moving surfaces closest point method grid based particle method radial basis functions finite differences (RBF-FD) least-squares method The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented. NSERC Canada Grant (RGPIN 2016-04361), Hong Kong Research Grant Council GRF Grant, Hong Kong Baptist University FRG Grant http://hdl.handle.net/20.500.11824/909
title A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
spellingShingle A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
Petras, A.
partial differential equations on moving surfaces
closest point method
grid based particle method
radial basis functions finite differences (RBF-FD)
least-squares method
title_short A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
title_full A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
title_fullStr A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
title_full_unstemmed A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
title_sort A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
author Petras, A.
author_facet Petras, A.
Ling, L.
Piret, C.
Ruuth, S.J.
author_role author
author2 Ling, L.
Piret, C.
Ruuth, S.J.
author2_role author
author
author
topic partial differential equations on moving surfaces
closest point method
grid based particle method
radial basis functions finite differences (RBF-FD)
least-squares method
topic_facet partial differential equations on moving surfaces
closest point method
grid based particle method
radial basis functions finite differences (RBF-FD)
least-squares method
description The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.
publishDate 2018
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status_str publishedVersion
url http://hdl.handle.net/20.500.11824/909
eu_rights_str_mv openAccess
institution Basque Center for Applied Mathematics (BCAM)
collection BIRD. BCAM's Institutional Repository Data
reponame_str BIRD. BCAM's Institutional Repository Data
instname_str Basque Center for Applied Mathematics (BCAM)
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publishDateSort 2018
author_browse Ling, L.
Petras, A.
Piret, C.
Ruuth, S.J.
score 6,924472

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