Issue |
ESAIM: M2AN
Volume 36, Number 1, January/February 2002
|
|
---|---|---|
Page(s) | 55 - 68 | |
DOI | https://doi.org/10.1051/m2an:2002003 | |
Published online | 15 April 2002 |
Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
1
Instituto de Ciencias, Univ. Nac. Gral. Sarmiento, J.M. Gutierrez entre Verdi y J.L. Suarez (1613), Los Polvorines, Buenos Aires, Argentina. gacosta@ungs.edu.ar.
2
Departamento de Matemática, FCEyN, UBA (1428), Buenos Aires, Argentina. jfbonder@dm.uba.ar. andjrossi@dm.uba.ar.
3
Universidad de San Andrés, Vito Dumas 284 (1644), Victoria, Buenos Aires, Argentina. pgroisman@udesa.edu.ar.
Received:
23
April
2001
Revised:
12
October
2001
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 ,
on ∂Ω x (0,T), where
Ω is a bounded smooth domain in
. 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.
Mathematics Subject Classification: 65M60 / 65M20 / 35K60 / 35B40
Key words: Blow-up / parabolic equations / semi-discretization in space / asymptotic behavior / non-linear boundary conditions.
© EDP Sciences, SMAI, 2002
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