Summary: This paper concerns the study of the numerical approximation for the following initial-boundary value problem
\[ \begin{cases} u_t- u_{xx}= f(u), \quad &x(0,1), \;t \in(0,T), \\ u(0,t)= 0, &u_x(1,t)= 0,\;t \in(0,T), \\ u(x,0)= u_0(x), &x \in[0,1], \end{cases} \]
where \(f(s)\) is a positive, increasing, \(C^1\) convex function for the nonnegative values of \(s, f(0)>0\), \(\int^\infty\frac{ds}{f(s)}<\infty,\) \(u_0\in C^1([0,1])\), \(u_0(0)=0\), \(u_0'(1) = 0\). We find some conditions under which the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove the convergence of the semidiscrete blow-up time to the theoretical one. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis.