Diabate Nabongo and Théodore K. Boni
abstract:
This paper concerns the study of the numerical approximation for the following
boundary value problem:
$$ \begin{cases}
u_t(x,t)-u_{xx}(x,t)=-u^{-p}(x,t), & 0<x<1,\;\; t>0, \\ u(0,t)=1, \;\; u(1,t)=1,
& t>0, \\ u(x,0)=u_{0}(x), & 0\leq x\leq1,
\end{cases} $$
where $p>0$. We obtain 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 establish the convergence of the
semidiscrete quenching time and construct two discrete forms of the above
problem which allow us to obtain some lower bounds of the numerical quenching
time. Finally, we give some numerical experiments to illustrate our theoretical
analysis.
Mathematics Subject Classification: 5K55, 35B40, 65M06
Key words and phrases: Semidiscretization, discretization, semilinear parabolic equation, semidiscrete quenching time, convergence