Abstract:
In this paper, we study the asymptotic behaviour of a semidiscrete numerical approximation for Ut = Uxx + up in a bounded interval, (0, 1), with Dirichlet boundary conditions. We focus in the behaviour of blowing up solutions. We find that the blow-up rate for the numerical scheme is the same as for the continuous problem. Also we find the blow-up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow-up set of the continuous problem when the mesh parameter is small enough. © 2001 Elsevier Science B.V. All rights reserved.
Registro:
Documento: |
Artículo
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Título: | Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions |
Autor: | Groisman, P.; Rossi, J.D. |
Filiación: | Departamento de Matemática, F.C.E y N., UBA, 1428 Buenos Aires, Argentina
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Palabras clave: | Asymptotic behaviour; Blow-up; Semidiscretization in space; Semilinear parabolic equations; Asymptotic stability; Boundary conditions; Semidiscrete numerical approximation; Approximation theory; mathematical analysis |
Año: | 2001
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Volumen: | 135
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Número: | 1
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Página de inicio: | 135
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Página de fin: | 155
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DOI: |
http://dx.doi.org/10.1016/S0377-0427(00)00571-9 |
Título revista: | Journal of Computational and Applied Mathematics
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Título revista abreviado: | J. Comput. Appl. Math.
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ISSN: | 03770427
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03770427_v135_n1_p135_Groisman |
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
- Ball, J., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations (1977) Quart. J. Math. Oxford, 28, pp. 473-486
- Bandle, C., Brunner, H., Blow-up in diffusion equations: A survey (1998) J. Comput. Appl. Math, 97, pp. 3-22
- Bandle, C., Brunner, H., Numerical analysis of semilinear parabolic problems with blow-up solutions (1994) Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid, 88, pp. 203-222
- 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
- Chen, X.Y., Asymptotic behaviours of blowing up solutions for finite difference analogue of u1 = uxx + u1+α (1986) J. Fac. Sci. Univ. Tokyo Sec IA Math, 33, pp. 541-574
- Chen, X.Y., Matano, H., Convergence, asymptotic periodicity and finite point blow up in one-dimensional semilinear heat equations (1989) J. Differential Equations, 78, pp. 160-190
- Ciarlet, P., (1978) The Finite Element Method for Elliptic Problems, , North-Holland, Amsterdam
- Cortázar, C., del Pino, M., Elgueta, M., The problem of uniqueness of the limit in a semilinear heat equation (1999) Comm. Partial Differential Equations, 24 (11-12), pp. 2147-2172
- Friedman, A., Mc Leod, J.B., Blow up of positive solutions of semilinear heat equations (1985) Indiana Univ. Math. J, 34, pp. 425-447
- Giga, Y., Kohn, R.V., Nondegeneracy of blow up for semilinear heat equations (1989) Comm. Pure Appl. Math, 42, pp. 845-884
- Henry, D., (1981) Geometric Theory of Semilinear Parabolic Equation, Lecture Notes in Mathematics, 840. , Springer, Berlin
- Herrero, M.A., Velazquez, J.J.L., Flat blow up in one-dimensional, semilinear parabolic problems (1992) Differential Integral Equations, 5 (5), pp. 973-997
- Herrero, M.A., Velazquez, J.J.L., Generic behaviour of one-dimensional blow up patterns (1992) Ann. Scuola Norm. Sup. Di Pisa, 19 (3), pp. 381-950
- Le Roux, M.N., Semidiscretizations in time of nonlinear parabolic equations with blow-up of the solutions (1994) SIAM J. Numer. Anal, 31, pp. 170-195
- Merle, F., Solution of a nonlinear heat equation with a arbitrarily given blow-up points (1992) Comm. Pure Appl. Math, 45, pp. 263-300
- Muller, C.E., Weissler, F.B., Single point blow up for a general semilinear heat equation (1983) Indiana Univ. Math. J, 34, pp. 881-913
- Nakagawa, T., Blowing up of a finite difference solution to ut = Uxx + u2 (1976) Appl. Math. Optim, 2, pp. 337-350
- Nakagawa, T., Ushijima, T., Finite element analysis of the semilinear heat equation of blow-up type (1977) Topics in Numerical Analysis, 3, pp. 275-291. , J.J.H. Miller (Ed.), Academic Press, London
- Pao, C.V., (1992) Nonlinear Parabolic and Elliptic Equations, , Plenum Press, New York
- Samarski, A., Galacktionov, V.A., Kurdyunov, S.P., Mikailov, A.P., (1995) Blow-up in Quasilinear Parabolic Equations, , Walter de Gruyter, Berlin
- Weissler, F.B., Single point blow up of semilinear initial boundary value problems (1984) J. Differential Equations, 55, pp. 204-224
Citas:
---------- APA ----------
Groisman, P. & Rossi, J.D.
(2001)
. Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions. Journal of Computational and Applied Mathematics, 135(1), 135-155.
http://dx.doi.org/10.1016/S0377-0427(00)00571-9---------- CHICAGO ----------
Groisman, P., Rossi, J.D.
"Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions"
. Journal of Computational and Applied Mathematics 135, no. 1
(2001) : 135-155.
http://dx.doi.org/10.1016/S0377-0427(00)00571-9---------- MLA ----------
Groisman, P., Rossi, J.D.
"Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions"
. Journal of Computational and Applied Mathematics, vol. 135, no. 1, 2001, pp. 135-155.
http://dx.doi.org/10.1016/S0377-0427(00)00571-9---------- VANCOUVER ----------
Groisman, P., Rossi, J.D. Asymptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions. J. Comput. Appl. Math. 2001;135(1):135-155.
http://dx.doi.org/10.1016/S0377-0427(00)00571-9