Numerical quenching of a heat equation with nonlinear boundary conditions

Abstract

In this paper, we study the quenching behavior of semidiscretizations of the heat equation with nonlinear boundary conditions. We obtain some conditions under which the positive solution of the semidiscrete problem 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. Finally we give some numerical results to illustrate our analysis.

Quenching for discretizations of a semilinear parabolic equation with nonlinear boundary outflux

“…We also prove that, under suitable assumptions on the initial datum, the semidiscrete quenching time converges to the theoretical one when the mesh size goes to zero. Our work was motived by the papers in [9,1,16,2,10,4], where the authors have used semidiscrete forms for some parabolic equations to study the phenomenon of blow-up (we say that a solution blows up in a finite time if it reaches the value infinity in a finite time).…”