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

Gabriel Acosta; Julián Fernández Bonder; Pablo Groisman; Julio Daniel Rossi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 1, page 55-68
  • ISSN: 0764-583X

Abstract

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We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations u t = Δ u , v t = Δ v in Ω × ( 0 , T ) ; fully coupled by the boundary conditions ∂ u ∂ η = u p 11 v p 12 , ∂ v ∂ η = u p 21 v p 22 on ∂ Ω × ( 0 , T ) , where Ω is a bounded smooth domain in ℝ d . 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 p 11 > 1 and p 21 < 2 ( p 11 - 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.

How to cite

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Acosta, Gabriel, et al. "Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.1 (2002): 55-68. <http://eudml.org/doc/245881>.

@article{Acosta2002,
abstract = {We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations $u_t = \Delta u$, $v_t = \Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac\{\partial u\}\{\partial \eta \} = u^\{p_\{11\}\}v^\{p_\{12\}\}$, $\frac\{\partial v\}\{\partial \eta \} = u^\{p_\{21\}\}v^\{p_\{22\}\}$ on $\partial \Omega \times (0,T)$, where $\Omega $ is a bounded smooth domain in $\mathbb \{R\}^d$. 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 $p_\{11\}&gt; 1$ and $p_\{21\} &lt; 2(p_\{11\}-1)$, 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, Gabriel, Bonder, Julián Fernández, Groisman, Pablo, Rossi, Julio Daniel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {blow-up; parabolic equations; semi-discretization in space; asymptotic behavior; non-linear boundary conditions; nonlinear boundary conditions; convergence},
language = {eng},
number = {1},
pages = {55-68},
publisher = {EDP-Sciences},
title = {Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions},
url = {http://eudml.org/doc/245881},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Acosta, Gabriel
AU - Bonder, Julián Fernández
AU - Groisman, Pablo
AU - Rossi, Julio Daniel
TI - Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 1
SP - 55
EP - 68
AB - We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations $u_t = \Delta u$, $v_t = \Delta v$ in $\Omega \times (0,T)$; fully coupled by the boundary conditions $\frac{\partial u}{\partial \eta } = u^{p_{11}}v^{p_{12}}$, $\frac{\partial v}{\partial \eta } = u^{p_{21}}v^{p_{22}}$ on $\partial \Omega \times (0,T)$, where $\Omega $ is a bounded smooth domain in $\mathbb {R}^d$. 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 $p_{11}&gt; 1$ and $p_{21} &lt; 2(p_{11}-1)$, 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.
LA - eng
KW - blow-up; parabolic equations; semi-discretization in space; asymptotic behavior; non-linear boundary conditions; nonlinear boundary conditions; convergence
UR - http://eudml.org/doc/245881
ER -

References

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