Continuity of the numerical quenching time in a semilinear heat equation. (English) Zbl 1524.35084
Summary: This paper concerns the study of the numerical approximation for the following initial-boundary value problem:
\[
\begin{gathered}
u_t = u_{xx}- u^{-p},\quad x\in (0, 1),\quad t\in (0, T_q),\\
u_x (0, t) = 0,\quad u_x (1, t) = 0,\quad t\in (0, T_q),\\
u(x, 0) = u_0 (x) > 0,\quad x\in [0, 1].
\end{gathered}
\]
Under some assumptions, we prove that the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its numerical quenching time. We also prove the continuity of the quenching time as a function of the initial datum. Finally, we give some numerical experiments to illustrate our analysis.
MSC:
35B40 | Asymptotic behavior of solutions to PDEs |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
35K58 | Semilinear parabolic equations |