Abstract:
The equation u t =Δu+u p with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.
Registro:
Documento: |
Artículo
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Título: | Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions |
Autor: | Groisman, P. |
Filiación: | Universidad de Buenos Aires, Argentina Instituto de Cálculo, Fac. Ciencias Exactas y Naturales, Pabellón II Ciudad Universitaria (1428), Buenos Aires, Argentina
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Palabras clave: | Adaptive numerical scheme; Asymptotic behavior; Blow-up; Adaptive numerical scheme; Asymptotic behavior; Blow-up; Approximation theory; Asymptotic stability; Problem solving; Boundary conditions |
Año: | 2005
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Volumen: | 76
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Número: | 3-4
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Página de inicio: | 325
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Página de fin: | 352
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DOI: |
http://dx.doi.org/10.1007/s00607-005-0136-0 |
Título revista: | Computing (Vienna/New York)
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Título revista abreviado: | Comput Vienna New York
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ISSN: | 0010485X
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CODEN: | CMPTA
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0010485X_v76_n3-4_p325_Groisman |
Referencias:
- Abia, L.M., López-Marcos, J.C., Martínez, J., Blow-up for semidiscretizations of reaction-diffusion equations (1996) Appl. Numer. Math., 20 (1-2), pp. 145-156
- (1995) Workshop on the Method of Lines for Time-dependent Problems, , Lexington, KY
- Abia, L.M., López-Marcos, J.C., Martínez, J., On the blow-up time convergence of semidiscretizations of reaction-diffusion equations (1998) Appl. Numer. Math., 26 (4), pp. 399-414
- Abia, L.M., López-Marcos, J.C., Martínez, J., The Euler method in the numerical integration of reaction-diffusion problems with blow-up (2001) Appl. Numer. Math., 38 (3), pp. 287-313
- Acosta, G., Durán, R.G., Rossi, J.D., An adaptive time step procedure for a parabolic problem with blow-up (2002) Computing, 68 (4), pp. 343-373
- Bandle, C., Brunner, H., Numerical analysis of semilinear parabolic problems with blow-up solutions (1994) Rev. Real Acad. Cienc. Exact. Fí S. Natur. Madrid, 88 (2-3), pp. 203-222
- Bandle, C., Brunner, H., Blow-up in diffusion equations: A survey (1998) J. Comput. Appl. Math., 97 (1-2), pp. 3-22
- Berger, M., Kohn, R.V., A rescaling algorithm for the numerical calculation of blowing-up solutions (1988) Comm. Pure Appl. Math., 41 (6), pp. 841-863
- Budd, C.J., Chen, S., Russell, R.D., New self-similar solutions of the nonlinear Schrödinger equation with moving mesh computations (1999) J. Comput. Phys., 152 (2), pp. 756-789
- Budd, C.J., Collins, G.J., An invariant moving mesh scheme for the nonlinear diffusion equation (1998) Proc. Int. Centre for Math. Sci. Conf. on Grid Adaptation in Computational PDEs: Theory and Applications, 26, pp. 23-39. , Edinburgh, 1996
- Budd, C.J., Huang, W., Russell, R.D., Moving mesh methods for problems with blow-up (1996) SIAM J. Sci. Comput., 17 (2), pp. 305-327
- Chen, Y.G., Asymptotic behaviours of blowing-up solutions for finite difference analogue of u t = u xx + u 1+α (1986) J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33 (3), pp. 541-574
- Ciarlet, P.G., The finite element method for elliptic problems (1978) Studies in Mathematics and its Applications, 4. , Amsterdam: North-Holland
- Cortázar, C., Del Pino, M., Elgueta, M., The problem of uniqueness of the limit in a semilinear heat equation (1999) Comm. Partial Differ. Eq., 24 (11-12), pp. 2147-2172
- Duran, R.G., Etcheverry, J.I., Rossi, J.D., Numerical approximation of a parabolic problem with a nonlinear boundary condition (1998) Discrete Contin. Dyn. Sys., 4 (3), pp. 497-506
- Giga, Y., Kohn, R.V., Characterizing blow-up using similarity variables (1987) Indiana Univ. Math. J., 36 (1), pp. 1-40
- Giga, Y., Kohn, R.V., Nondegeneracy of blow-up for semilinear heat equations (1989) Comm. Pure Appl. Math., 42 (6), pp. 845-884
- Groisman, P., Rossi, J.D., Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions (2001) J. Comput. Appl. Math., 135 (1), pp. 135-155
- Harary, F., (1969) Graph Theory, , Reading, MA/Menlo Park, CA/London: Addison-Wesley
- Huang, W., Ren, Y., Russell, R.D., Moving mesh partial differential equations (MMPDES) based on the equidistribution principle (1994) SIAM J. Numer. Anal., 31 (3), pp. 709-730
- Le Roux, M.-N., Semidiscretization in time of nonlinear parabolic equations with blow-up of the solution (1994) SIAM J. Numer. Anal., 31 (1), pp. 170-195
- Nakagawa, T., Blowing up of a finite difference solution to u t = u xx + u 2 (1975) Appl. Math. Optim., 2 (4), pp. 337-350
- Nakagawa, T., Ushijima, T., Finite element analysis of the semi-linear heat equation of blow-up type (1977) Topics in Numerical Analysis, III, pp. 275-291. , J. J. H. Miller (ed.). Published for the Royal Irish Academy by: London New York: Academic Press
- Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Blow-up in quasilinear parabolic equations (1995) Gruyter Expositions in Mathematics, 19. , Berlin: de Gruyter. Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors
- Stuart, A.M., Floater, M.S., On the computation of blow-up (1990) Eur. J. Appl. Math., 1 (1), pp. 47-71. , Europ. vol. 19
- Ushijima, T.K., On the approximation of blow-up time for solutions of nonlinear parabolic equations (2000) Publ. Res. Inst. Math. Sci., 36 (5), pp. 613-640
- Velázquez, J.J.L., Classification of singularities for blowing up solutions in higher dimensions (1993) Trans. Amer. Math. Soc., 338 (1), pp. 441-464
Citas:
---------- APA ----------
(2005)
. Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions. Computing (Vienna/New York), 76(3-4), 325-352.
http://dx.doi.org/10.1007/s00607-005-0136-0---------- CHICAGO ----------
Groisman, P.
"Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions"
. Computing (Vienna/New York) 76, no. 3-4
(2005) : 325-352.
http://dx.doi.org/10.1007/s00607-005-0136-0---------- MLA ----------
Groisman, P.
"Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions"
. Computing (Vienna/New York), vol. 76, no. 3-4, 2005, pp. 325-352.
http://dx.doi.org/10.1007/s00607-005-0136-0---------- VANCOUVER ----------
Groisman, P. Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions. Comput Vienna New York. 2005;76(3-4):325-352.
http://dx.doi.org/10.1007/s00607-005-0136-0