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.
Blow-up of solutions of nonlinear parabolic inequalities