NUMERICAL QUENCHING SOLUTIONS OF LOCALIZED SEMILINEAR PARABOLIC EQUATION

This paper concerns the study of the numerical approximation for the following initial-boundary value problem: {u(t) (x,t) = u(xx) (x,t) + epsilon (1-u(0,t))(-p), (x,t) epsilon (-l,l) x (0,T), u(-l,t) = 0, u(l,t) = 0, t epsilon (0, T), u(x,0) = u(0)(x) >= 0, x epsilon (-l,l), where p > 1, 1 = 1/2 and epsilon > 0. Under some assumptions, we prove that the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenching time. We also show that the semidiscrete quenching time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.