Quenching for discretization of a nonlinear diffusion equation with singular boundary flux
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In this paper, we study the discrete approximation for the following nonlinear diffusion equation with nonlinear source and singular boundary flux $$\\ \left\{% \begin{array}{ll} \hbox{$\dfrac{\partial A(u)}{\partial t}=u_{xx} + (1-u)^{-\alpha}, \quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)= 0, \quad u_{x}(1,t)=-B(u(1,t)),\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x),\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ with $ \alpha > 0. $ We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.
Abstract
In this paper, we study the discrete approximation for the following nonlinear diffusion equation with nonlinear source and singular boundary flux $$\\ \left\{% \begin{array}{ll} \hbox{$\dfrac{\partial A(u)}{\partial t}=u_{xx} + (1-u)^{-\alpha}, \quad 0<x<1,\; t>0$,} \\ \hbox{$u_{x}(0,t)= 0, \quad u_{x}(1,t)=-B(u(1,t)),\quad t>0$,} \\ \hbox{$u(x,0)=u_{0}(x),\quad 0\leq x \leq 1$,} \\ \end{array}% \right. $$ with $ \alpha > 0. $ We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.
참고문헌 (8)
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