This paper concerns the study of the numerical approximation for the following boundary value problem: $$\left\{\begin{array}{cccc}{u}_t(x,t)-u_{xx}(x,t)=-u^{-p}(x,t),\hfill & 0<x<1,t>0,u_x(0,t)=0,\hfill & u(1,t)=1,t>0,u(x,0)=u_0\left(x\right)>0,& 0\le x\le 1,\end{array}\right.$$ where $p>0$. 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.

In this paper, we consider the following initial-boundary value problem $$\left\}\begin{array}{c}{u}_{tt}(x,t)=\epsilon Lu(x,t)+f\left(u(x,t)\right)\text{in}\Omega \times (0,T),\hfill \\ u(x,t)=0\text{on}\partial \Omega \times (0,T),\hfill \\ u(x,0)=0\text{in}\Omega ,\hfill \\ u_t(x,0)=0\text{in}\Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded domain in ${ℝ}^N$ with smooth boundary $\partial \Omega$, $L$ is an elliptic operator, $\epsilon$ is a positive parameter, $f\left(s\right)$ is a positive, increasing, convex function for $s\in (-\infty ,b)$, ${lim}_{s\to b}f\left(s\right)=\infty$ and ${\int }_0^b\frac{ds}{f\left(s\right)}<\infty$ with $b=const>0$. Under some assumptions, we show that the solution of the above problem quenches in a finite time and its quenching time goes to that of the solution of the following differential equation $$\left\}\begin{array}{cc}{\alpha }^{\text{'}\text{'}}\left(t\right)=f\left(\alpha \left(t\right)\right),\hfill & t>0,\hfill \\ \alpha \left(0\right)=0,{\alpha }^{\text{'}}\left(0\right)=0,\hfill \end{array}\right.$$ as $\epsilon$ goes to zero. We also show that the above result remains...