Numerical quenching for a nonlinear diffusion equation with a singular boundary condition

Abstract

This paper concerns the study of the numerical approximation for the following boundary value problem $$ \left\{ \begin{array}{ll} \hbox{$(u^{m})_{t}=u_{xx}$, $<x<1$,\; $t>0$,} \\ \hbox{$u_{x}(0,t)=0$,\quad $u_{x}(1,t)=-u^{-\beta}(1,t)$,\quad $t>0$,} \\ \hbox{$u(x,0)=u_{0}(x)>0$,\quad $0\leq x\leq 1$,} \\ \end{array} \right.$$ where $m\geq1$, $\beta>0$. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time. Finally, we give some numerical experiments to illustrate our analysis.