Artículo
Acosta, G.; Durán, R.G.; Rossi, J.D. "An adaptive time step procedure for a parabolic problem with blow-up" (2002) Computing (Vienna/New York). 68(4):343-373
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Abstract:
In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blowup of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme.
Registro:
| Documento: | Artículo |
| Título: | An adaptive time step procedure for a parabolic problem with blow-up |
| Autor: | Acosta, G.; Durán, R.G.; Rossi, J.D. |
| Filiación: | Instituto de Ciencias, Universidad de General Sarmiento J.M.Gutierrez, entre Verdi y J.L. Suarez (1613), Buenos Aires, Argentina Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (1428), Buenos Aires, Argentina |
| Palabras clave: | Adaptivity; Blow up; Nonlinear boundary conditions; Numerical approximations; Approximation theory; Boundary conditions; Differential equations; Nonlinear systems; Adaptive time step procedures; Parabolic problems; Problem solving |
| Año: | 2002 |
| Volumen: | 68 |
| Número: | 4 |
| Página de inicio: | 343 |
| Página de fin: | 373 |
| DOI: | http://dx.doi.org/10.1007/s00607-002-1449-x |
| 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_v68_n4_p343_Acosta |
Referencias:
- Abia, L.M., Lopez-Marcos, J.C., Martinez, J., Blow-up for semidiscretizations of reaction diffusion equations (1996) Appl. Numer. Math., 20, pp. 145-156
- Abia, L.M., Lopez-Marcos, J.C., Martinez, J., On the blow-up time convergence of semidiscretizations of reaction diffusion equations (1998) Appl. Numer. Math., 26, pp. 399-414
- Berger, M., Kohn, R.V., A rescaling algorithm for the numerical calculation of blowing up solutions (1988) Comm. Pure Appl. Math., 41, pp. 841-863
- 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
- Durán, R.G., Etcheverry, J.I., Rossi, J.D., Numerical approximation of a parabolic problem with a nonlinear boundary condition (1998) Discr. Cont. Dyn. Sys., 4 (3), pp. 497-506
- López Gómez, J., Márquez, V., Wolanski, N., Blow-up results and localization of blow-up points for the heat equation with a nonlinear boundary condition (1991) J. Diff. Eq., 92 (2), pp. 384-401
- Pao, C.V., (1992) Nonlinear parabolic and elliptic equations, , Plenum Press
- Rial, D.F., Rossi, J.D., Blow-up results and localization of blow-up points in an n-dimensional smooth domain (1997) Duke Math. Jour., 88 (2), pp. 391-405
- Sanz-Serna, J.M., Verwer, J.G., A study of the recursion yn+1 = yn + τynm (1986) J. Math. Anal. Appl., 116, pp. 456-464
- Walter, W., On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition (1975) SIAM J. Math. Anal., 6 (1), pp. 85-90
Citas:
---------- APA ----------
Acosta, G., Durán, R.G. & Rossi, J.D. (2002) . An adaptive time step procedure for a parabolic problem with blow-up. Computing (Vienna/New York), 68(4), 343-373.http://dx.doi.org/10.1007/s00607-002-1449-x
---------- CHICAGO ----------
Acosta, G., Durán, R.G., Rossi, J.D. "An adaptive time step procedure for a parabolic problem with blow-up" . Computing (Vienna/New York) 68, no. 4 (2002) : 343-373.http://dx.doi.org/10.1007/s00607-002-1449-x
---------- MLA ----------
Acosta, G., Durán, R.G., Rossi, J.D. "An adaptive time step procedure for a parabolic problem with blow-up" . Computing (Vienna/New York), vol. 68, no. 4, 2002, pp. 343-373.http://dx.doi.org/10.1007/s00607-002-1449-x
---------- VANCOUVER ----------
Acosta, G., Durán, R.G., Rossi, J.D. An adaptive time step procedure for a parabolic problem with blow-up. Comput Vienna New York. 2002;68(4):343-373.http://dx.doi.org/10.1007/s00607-002-1449-x
