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 (2010)

  • 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 ut = Δu, vt = Δv in Ω x (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 x (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 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.

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 36.1 (2010): 55-68. <http://eudml.org/doc/194096>.

@article{Acosta2010,
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. },
author = {Acosta, Gabriel, Bonder, Julián Fernández, Groisman, Pablo, Rossi, Julio Daniel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Blow-up; parabolic equations; semi-discretization in space; asymptotic behavior; non-linear boundary conditions.; blow-up; asymptotic behavior; nonlinear boundary conditions; convergence},
language = {eng},
month = {3},
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/194096},
volume = {36},
year = {2010},
}

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
DA - 2010/3//
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 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.
LA - eng
KW - Blow-up; parabolic equations; semi-discretization in space; asymptotic behavior; non-linear boundary conditions.; blow-up; asymptotic behavior; nonlinear boundary conditions; convergence
UR - http://eudml.org/doc/194096
ER -

References

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  1. L.M. Abia, J.C. Lopez-Marcos and J. Martinez, Blow-up for semidiscretizations of reaction diffusion equations. Appl. Numer. Math.20 (1996) 145-156.  
  2. L.M. Abia, J.C. Lopez-Marcos and J. Martinez, On the blow-up time convergence of semidiscretizations of reaction diffusion equations. Appl. Numer. Math.26 (1998) 399-414.  
  3. G. Acosta, J. Fernández Bonder, P. Groisman and J.D. Rossi. Numerical approximation of a parabolic problem with nonlinear boundary condition in several space dimensions. Preprint.  
  4. H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations.72 (1988) 201-269.  
  5. C. Bandle and H. Brunner, Blow-up in diffusion equations: a survey. J. Comput. Appl. Math.97 (1998) 3-22.  
  6. M. Berger and R.V. Kohn, A rescaling algorithm for the numerical calculation of blowing up solution. Comm. Pure Appl. Math.41 (1988) 841-863.  
  7. C.J. Budd, W. Huang and R.D. Russell, Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput.17 (1996) 305-327.  
  8. Y.G. Chen, Asymptotic behaviours of blowing up solutions for finite difference analogue of u t = u x x + u 1 + α . J. Fac. Sci. Univ. Tokyo Sect. IA Math.33 (1986) 541-574.  
  9. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).  
  10. R.G. Durán, J.I. Etcheverry and J.D. Rossi, Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete Contin. Dyn. Syst.4 (1998) 497-506.  
  11. C.M. Elliot and A.M. Stuart, Global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal.30 (1993) 1622-1663.  
  12. J. Fernández Bonder and J.D. Rossi, Blow-up vs. spurious steady solutions. Proc. Amer. Math. Soc.129 (2001) 139-144.  
  13. A.R. Humphries, D.A. Jones and A.M. Stuart, Approximation of dissipative partial differential equations over long time intervals, in D.F. Griffiths et al., Eds., Numerical Analysis 1993. Proc. 15th Dundee Biennal Conf. on Numerical Analysis, June 29-July 2nd, 1993, University of Dundee, UK, in Pitman Res. Notes Math. Ser. 303, Longman Scientific & Technical, Harlow (1994) 180-207.  
  14. C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992).  
  15. J.P. Pinasco and J.D. Rossi, Simultaneousvs. non-simultaneous blow-up. N. Z. J. Math.29 (2000) 55-59.  
  16. J.D. Rossi, On existence and nonexistence in the large for an N-dimensional system of heat equations with nontrivial coupling at the boundary. N. Z. J. Math.26 (1997) 275-285.  
  17. A. Samarski, V.A. Galaktionov, S.P. Kurdyunov and A.P. Mikailov, Blow-up in QuasiLinear Parabolic Equations, in Walter de Gruyter, Ed., de Gruyter Expositions in Mathematics 19, Berlin (1995).  
  18. A.M. Stuart and A.R. Humphries, Dynamical systems and numerical analysis, in Cambridge Monographs on Applied and Computational Mathematics 2, Cambridge University Press, Cambridge (1998).  

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