Artículo

La versión final de este artículo es de uso interno de la institución.
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

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
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
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
Volumen:76
Número:3-4
Página de inicio:325
Página de fin:352
DOI: http://dx.doi.org/10.1007/s00607-005-0136-0
Título revista:Computing (Vienna/New York)
Título revista abreviado:Comput Vienna New York
ISSN:0010485X
CODEN:CMPTA
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