Quenching for semidiscretizations of a semilinear heat equation with potential and general nonlinearity

Abstract

This paper concerns the study of the numerical approximation for the following boundary value problem  \begin{equation*} \begin{cases} u_t(x,t)-u_{xx}(x,t)= -a(x)f(u(x,t)), & 0<x<1,\; t>0, \\ u_x(0,t)= 0, u(1,t)=0, & t>0, \\ u(x,0)= u_{0}(x)>0, & 0\leq x \leq 1, \\ \end{cases} \end{equation*} where \(f:(0,\infty)\rightarrow (0,\infty)\) is a \(C^{1}\) convex, nondecreasing function, \(\lim_{s\rightarrow 0^{+}}f(s)=\infty,\) \(\int_0^{\alpha}\tfrac{{\rm d}s}{f(s)}<\infty \) for any positive real \(\alpha.\) The initial datum \(u_{0}\in C^{2}([0,1]\) \(u'_{0}(0)=0\) and \(u'_{0}(1)=0\). The potential \(a\in C^{1}((0,1))\), \(a(x)>0,\) \(x \in (0,1),\) \(a'(0)=0,\) \(a'(1)=0.\) We find 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 prove that the semidiscrete quenching time converges to the real one when the mesh size goes to zero. A similar study has been also investigated taking a discrete form of the above problem. Finally, we give some numerical experiments to illustrate our analysis.