In this paper, we examine the blow-up approximation for the semilinear reaction-diffusion equation of the following: \( \left\{ \begin{array}{ll} \displaystyle u_t(x,t)=u_{xx}(x,t)+u^{p}(x,t), & x\in (0,1), \ t\in (0,T^{*}), \\ \displaystyle u(0,t)=0, \ \ u(1,t)=0, & t\in (0,T^*), \\ \displaystyle u(x,0)=u_{0}(x)\ge 0, & x\in [0,1], \end{array} \right. \) where \(p > 1\) . By developing an appropriate semi-discrete finite difference system and incorporating the concavity method alongside a first-order inequality technique in discrete form, we derive sufficient conditions for the finite-time blow-up of solutions and establish upper bounds for the blow-up time. Furthermore, utilizing the embedding inequality and constructing suitable auxiliary functions in discrete form, we establish a lower bound for the blow-up time within the discrete scheme. Additionally, we prove the convergence of the finite difference system and the blow-up time. Lastly, numerical experiments are presented to substantiate the accuracy of the results obtained in this paper.
