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

doi:10.1051/m2an:2002003

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

ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 55-68

Résumé

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 Zbl

DOI : 10.1051/m2an:2002003

Classification : 65M60, 65M20, 35K60, 35B40
Keywords: blow-up, parabolic equations, semi-discretization in space, asymptotic behavior, non-linear boundary conditions

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     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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {55--68},
     year = {2002},
     publisher = {EDP Sciences},
     volume = {36},
     number = {1},
     doi = {10.1051/m2an:2002003},
     mrnumber = {1916292},
     zbl = {1003.65097},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an:2002003/}
}

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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: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 1, pp. 55-68. doi: 10.1051/m2an:2002003

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