An adaptive time step procedure for a parabolic problem with blow-up
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 th...
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2002
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paper:paper_0010485X_v68_n4_p343_Acosta2023-06-08T14:34:29Z An adaptive time step procedure for a parabolic problem with blow-up Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Rossi, Julio Daniel Adaptivity Blow up Nonlinear boundary conditions Numerical approximations Approximation theory Boundary conditions Differential equations Nonlinear systems Adaptive time step procedures Parabolic problems Problem solving 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. Fil:Acosta, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Durán, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2002 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0010485X_v68_n4_p343_Acosta http://hdl.handle.net/20.500.12110/paper_0010485X_v68_n4_p343_Acosta |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Adaptivity Blow up Nonlinear boundary conditions Numerical approximations Approximation theory Boundary conditions Differential equations Nonlinear systems Adaptive time step procedures Parabolic problems Problem solving |
| spellingShingle |
Adaptivity Blow up Nonlinear boundary conditions Numerical approximations Approximation theory Boundary conditions Differential equations Nonlinear systems Adaptive time step procedures Parabolic problems Problem solving Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Rossi, Julio Daniel An adaptive time step procedure for a parabolic problem with blow-up |
| topic_facet |
Adaptivity Blow up Nonlinear boundary conditions Numerical approximations Approximation theory Boundary conditions Differential equations Nonlinear systems Adaptive time step procedures Parabolic problems Problem solving |
| description |
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. |
| author |
Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Rossi, Julio Daniel |
| author_facet |
Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Rossi, Julio Daniel |
| author_sort |
Acosta Rodriguez, Gabriel |
| title |
An adaptive time step procedure for a parabolic problem with blow-up |
| title_short |
An adaptive time step procedure for a parabolic problem with blow-up |
| title_full |
An adaptive time step procedure for a parabolic problem with blow-up |
| title_fullStr |
An adaptive time step procedure for a parabolic problem with blow-up |
| title_full_unstemmed |
An adaptive time step procedure for a parabolic problem with blow-up |
| title_sort |
adaptive time step procedure for a parabolic problem with blow-up |
| publishDate |
2002 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0010485X_v68_n4_p343_Acosta http://hdl.handle.net/20.500.12110/paper_0010485X_v68_n4_p343_Acosta |
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