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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Detalles Bibliográficos
Autores principales:	Acosta, G., Durán, R.G., Rossi, J.D.
Formato:	JOUR
Materias:	Adaptivity Blow up Nonlinear boundary conditions Numerical approximations Approximation theory Boundary conditions Differential equations Nonlinear systems Adaptive time step procedures Parabolic problems Problem solving
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_0010485X_v68_n4_p343_Acosta
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_0010485X_v68_n4_p343_Acosta
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spelling	todo:paper_0010485X_v68_n4_p343_Acosta2023-10-03T14:09:14Z An adaptive time step procedure for a parabolic problem with blow-up Acosta, G. Durán, R.G. Rossi, J.D. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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, G. Durán, R.G. Rossi, J.D. 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.
format	JOUR
author	Acosta, G. Durán, R.G. Rossi, J.D.
author_facet	Acosta, G. Durán, R.G. Rossi, J.D.
author_sort	Acosta, G.
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
url	http://hdl.handle.net/20.500.12110/paper_0010485X_v68_n4_p343_Acosta
work_keys_str_mv	AT acostag anadaptivetimestepprocedureforaparabolicproblemwithblowup AT duranrg anadaptivetimestepprocedureforaparabolicproblemwithblowup AT rossijd anadaptivetimestepprocedureforaparabolicproblemwithblowup AT acostag adaptivetimestepprocedureforaparabolicproblemwithblowup AT duranrg adaptivetimestepprocedureforaparabolicproblemwithblowup AT rossijd adaptivetimestepprocedureforaparabolicproblemwithblowup
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An adaptive time step procedure for a parabolic problem with blow-up

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