Solutions of initial value problems do not always exist globally in time and may become unbounded in a finite time. Such a phenomenon is often referred to as blow-up and the finite time is often called the blow-up time. To construct a numerical solution which also blows up in a finite time, adaptive temporal strategies were considered to be necessary. The approximate blow-up times, however, are usually defined by infinite sums, which cannot be carried out in real computation. As a result, we reconsider the schemes with uniform time meshes and propose an algorithm to compute an approximate blow-up time, which can be achieved concretely with rigorous convergence proofs. We will introduce the idea via a simple ODE blow-up problem and several applications for the computation of certain blow-up behaviors to PDE problems will be explored, too. Furthermore, such an idea is also applicable to the algorithms with adaptive temporal grids. Remarks on this issue will also be addressed.