A second-order in time finite-difference scheme using a modified predictor-corrector method is proposed for the numerical solution of the generalized Burgers-Fisher equation. The method introduced, which, in contrast to the classical predictor-corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.