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: 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/

References

[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. | Zbl 0857.65096

[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. | Zbl 0929.65070

[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. | Zbl 0997.35025

[4] H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations. 72 (1988) 201-269. | Zbl 0658.34011

[5] C. Bandle and H. Brunner, Blow-up in diffusion equations: a survey. J. Comput. Appl. Math. 97 (1998) 3-22. | Zbl 0932.65098

[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. | Zbl 0652.65070

[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. | Zbl 0860.35050

[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. | Zbl 0616.65098

[9] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

[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. | Zbl 0951.65088

[11] C.M. Elliot and A.M. Stuart, Global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30 (1993) 1622-1663. | Zbl 0792.65066

[12] J. Fernández Bonder and J.D. Rossi, Blow-up vs. spurious steady solutions. Proc. Amer. Math. Soc. 129 (2001) 139-144. | Zbl 0970.35003

[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. | Zbl 0795.65031

[14] C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992). | MR 1212084 | Zbl 0777.35001

[15] J.P. Pinasco and J.D. Rossi, Simultaneousvs. non-simultaneous blow-up. N. Z. J. Math. 29 (2000) 55-59. | Zbl 0951.35019

[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. | Zbl 0891.35053

[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). | Zbl 1020.35001

[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). | MR 1402909 | Zbl 0913.65068