Numerical blow-up for nonlinear diffusion equation with neumann boundary conditions

Volume 14, Issue 2, pp 80--88 http://dx.doi.org/10.22436/jnsa.014.02.03

Publication Date: July 03, 2020 Submission Date: February 25, 2020 Revision Date: March 03, 2020 Accteptance Date: April 12, 2020

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Authors

Ardjouma Ganon - Institut National Polytechnique Felix Houphouet-Boigny Yamoussoukro, BP 2444, Cote d'Ivoire. Manin Mathurin Taha - Institut National Polytechnique Felix Houphouet-Boigny Yamoussoukro, BP 2444, Cote d'Ivoire. N'guessan Koffi - UFR SED, Universite Alassane Ouattara de Bouake, 01 BP V 18 Bouake 01, Cote d'Ivoire. Augustin Kidjegbo Toure - UFR SED, Universite Alassane Ouattara de Bouake, 01 BP V 18 Bouake 01, Cote d'Ivoire.

Abstract

This work is concerned with the study of the numerical approximation for the nonlinear diffusion equation \( (u^{m})_t= u_{xx}, \ 0<x<1, \ t>0 \), under Neumann boundary conditions \( u_x(0,t)=0, \ u_x(1,t)=u^{\alpha}(1,t), \ t>0 \). First, we obtain a semidiscrete scheme by the finite differences method and prove the convergence of its solution to the continuous one. Then, we establish the numerical blow-up and the convergence of the numerical blow-up time to the theoretical one when the mesh size goes to zero. Finally, we illustrate our analysis with some numerical experiments.

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Keywords

• Nonlinear diffusion equation
• numerical blow-up
• arc length transformation
• Aitken \( \Delta^{2} \) method

MSC

•  35B44
•  35B51
•  35K20
•  65M06

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