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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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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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 |
work_keys_str_mv |
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_version_ |
1769810290308808704 |