This paper studies a finite difference approximation to the similinear heat equation (1) with special emphasis on the case when the exact solution blows up with the blowing-up timeT . The key results will be given in Propositions 1 and 2. Proposition 1 states the local convergence, i.e., the convergence of the proposed finite difference solution to the exact solution in any fixed time interval 0 t T, whereT <> . Proposition 2 states the convergence of the numerical blowing-up time to the exact oneT .