This paper concerns the study of the
numerical approximation for the following parabolic equations with a nonlinear convection term $$\\
\left\{%
\begin{array}{ll}
\hbox{$u_t(x,t)=u_{xx}(x,t)-g(u(x,t))u_{x}(x,t)+f(u(x,t)),\quad 0<x<1,\; t>0$,} \\
\hbox{$u_{x}(0,t)=0, \quad u_{x}(1,t)=0,\quad t>0$,} \\
\hbox{$u(x,0)=u_{0}(x) > 0,\quad 0\leq x \leq 1$,} \\
\end{array}%
\right.
$$
\newline
where $f:[0,+\infty)\rightarrow [0,+\infty)$ is $C^3$ convex,
nondecreasing function,\\ $g:[0,+\infty)\rightarrow [0,+\infty)$ is
$C^1$ convex, nondecreasing function,\newline
$\displaystyle\lim_{s\rightarrow +\infty}f(s)=+\infty$,
$\displaystyle\lim_{s\rightarrow +\infty}g(s)=+\infty$,
$\displaystyle\lim_{s\rightarrow +\infty}\frac{f(s)}{g(s)}=+\infty$\newline
and $\displaystyle\int^{+\infty}_{c}\frac{ds}{f(s)}<+\infty$ for
$c>0$. We obtain some conditions under which the solution of the
semidiscrete form of the above problem blows up in a finite time and
estimate its semidiscrete blow-up time. We also prove that the
semidiscrete blow-up time converges to the real one, when the mesh
size goes to zero. Finally, we give some numerical results to
illustrate ours analysis.
The boundary behavior of the unique solution to a singular Dirichlet problem
