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: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0764583X_v36_n1_p55_Acosta http://hdl.handle.net/20.500.12110/paper_0764583X_v36_n1_p55_Acosta |
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paper:paper_0764583X_v36_n1_p55_Acosta2023-06-08T15:45:45Z Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions Acosta Rodriguez, Gabriel Groisman, Pablo Jose Rossi, Julio Daniel 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 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0764583X_v36_n1_p55_Acosta 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) |
| 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 Rodriguez, Gabriel Groisman, Pablo Jose Rossi, Julio Daniel 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. |
| author |
Acosta Rodriguez, Gabriel Groisman, Pablo Jose Rossi, Julio Daniel |
| author_facet |
Acosta Rodriguez, Gabriel Groisman, Pablo Jose Rossi, Julio Daniel |
| author_sort |
Acosta Rodriguez, Gabriel |
| 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 |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0764583X_v36_n1_p55_Acosta http://hdl.handle.net/20.500.12110/paper_0764583X_v36_n1_p55_Acosta |
| work_keys_str_mv |
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1768544279878172672 |