Article contents
Simultaneous vs. non-simultaneous blow-up in numerical approximations of aparabolic system with non-linear boundary conditions
Published online by Cambridge University Press: 15 April 2002
Abstract
We study the asymptotic behavior of a semi-discrete numerical
approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (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 ∂Ω x (0,T), where
Ω 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 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.
Keywords
- Type
- Research Article
- Information
- ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 1 , January 2002 , pp. 55 - 68
- Copyright
- © EDP Sciences, SMAI, 2002
References
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