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

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

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

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 1, p. 55-68

Abstract

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 $Ω \times (0, T)$ ; fully coupled by the boundary conditions $\frac{\partial u}{\partial η} = u^{p_{11}} v^{p_{12}}$ , $\frac{\partial v}{\partial η} = u^{p_{21}} v^{p_{22}}$ on $\partial Ω \times (0, T)$ , 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.

MR 1916292 | Zbl 1003.65097

DOI : https://doi.org/10.1051/m2an:2002003
Classification: 65M60, 65M20, 35K60, 35B40
Keywords: blow-up, parabolic equations, semi-discretization in space, asymptotic behavior, non-linear boundary conditions

@article{M2AN_2002__36_1_55_0,
     author = {Acosta, Gabriel and Bonder, Juli\'an Fern\'andez and Groisman, Pablo and Rossi, Julio Daniel},
     title = {Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {1},
     year = {2002},
     pages = {55-68},
     doi = {10.1051/m2an:2002003},
     zbl = {1003.65097},
     mrnumber = {1916292},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_1_55_0}
}

Acosta, Gabriel; Bonder, Julián Fernández; Groisman, Pablo; Rossi, Julio Daniel. 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, Volume 36 (2002) no. 1, pp. 55-68. doi : 10.1051/m2an:2002003. http://www.numdam.org/item/M2AN_2002__36_1_55_0/

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