NUMERICAL BLOW-UP TIME FOR NONLINEAR PARABOLIC PROBLEMS
Keywords:
blow-up, nonlinear parabolic equation, finite difference scheme, numerical blow-up time.DOI:
https://doi.org/10.17654/0974324322028Abstract
In this paper, we analyze numerically some of the features of the blow-up phenomena arising from a nonlinear parabolic equation subject to nonlinear boundary conditions. More precisely, we study numerical approximations of solutions of the problem
$\left\{\begin{array}{l}(\log u(x, t))_t=u_{x x}(x, t)+u^{\beta-1}(x, t), \quad(x, t) \in(0,1) \times(0, T) \\ -u_x(0, t)+u^\alpha(0, t)=0, \quad t>0 \\ u_x(1, t)+u^\alpha(1, t)=0, \quad t>0 \\ u(x, 0)=u_0(x) \geq \gamma>0, \quad 0 \leq x \leq 1\end{array}\right.$
where $\beta \geq \alpha>1$. We obtain some conditions under which the solution of the semidiscrete form blows up in a finite time. We estimate its semidiscrete blow-up time and also establish the convergence of the semidiscrete blow-up time to the real one. Finally, we give some numerical experiments to illustrate our analysis.
Received: May 7, 2022
Accepted: June 22, 2022
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