Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω × (0, T); fully coupled by the boundary conditions ∂u/∂η = up11vp12, ∂v/∂η = up21vp22 on ∂Ω × (0, T), where Ω is a bounded smooth domain in ℝd. We focus in the existence or...

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Autores principales: Acosta, G., Bonder, J.F., Groisman, P., Rossi, J.D.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2002
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Acceso en línea:http://hdl.handle.net/20.500.12110/paper_0764583X_v36_n1_p55_Acosta
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spelling paperaa:paper_0764583X_v36_n1_p55_Acosta2023-06-12T16:48:15Z Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions Math. Model. Numer. Anal. 2002;36(1):55-68 Acosta, G. Bonder, J.F. Groisman, P. Rossi, J.D. Asymptotic behavior Blow-up Non-linear boundary conditions Parabolic equations Semi-discretization in space We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω × (0, T); fully coupled by the boundary conditions ∂u/∂η = up11vp12, ∂v/∂η = up21vp22 on ∂Ω × (0, T), where Ω is a bounded smooth domain in ℝd. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U, V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1), which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times. Fil:Acosta, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Groisman, P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2002 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_0764583X_v36_n1_p55_Acosta
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
language Inglés
orig_language_str_mv eng
topic Asymptotic behavior
Blow-up
Non-linear boundary conditions
Parabolic equations
Semi-discretization in space
spellingShingle Asymptotic behavior
Blow-up
Non-linear boundary conditions
Parabolic equations
Semi-discretization in space
Acosta, G.
Bonder, J.F.
Groisman, P.
Rossi, J.D.
Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
topic_facet Asymptotic behavior
Blow-up
Non-linear boundary conditions
Parabolic equations
Semi-discretization in space
description We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω × (0, T); fully coupled by the boundary conditions ∂u/∂η = up11vp12, ∂v/∂η = up21vp22 on ∂Ω × (0, T), where Ω is a bounded smooth domain in ℝd. We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U, V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1), which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times.
format Artículo
Artículo
publishedVersion
author Acosta, G.
Bonder, J.F.
Groisman, P.
Rossi, J.D.
author_facet Acosta, G.
Bonder, J.F.
Groisman, P.
Rossi, J.D.
author_sort Acosta, G.
title Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
title_short Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
title_full Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
title_fullStr Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
title_full_unstemmed Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
title_sort simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
publishDate 2002
url http://hdl.handle.net/20.500.12110/paper_0764583X_v36_n1_p55_Acosta
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