Quenching for semidiscretizations of a heat equation with a singular boundary condition

Abstract

We obtain some conditions under which the positive solution of the numerical approximation for the heat equation u_t(x, t)=u_xx(x, t), x∈(0, 1), t>0, with the singular boundary condition u_x(1, t)=−u^−β(1, t), where β>0 quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time and obtain some results on numerical quenching rate and set. Finally we give some numerical results to illustrate our analysis.