We propose a new path-integration calculation method to treat the time evolution of a wavefunction within the framework of the real-space finite-difference formalism, and also develop an effective scheme to compute the scattering wavefunction for an incident electron with arbitrary energy, in which an impulse wavefunction is adopted as an initial state of the time evolution. In this method, once the time evolution of the initial impulse wavefunction is calculated, all of the solutions in the scattering problem can be derived by means of Fourier analysis of the time-evolved wavefunction, which leads to a reduction of the calculation time. In order to test the applicability of our newly developed simulation procedures, we implemented simulations for the one-dimensional scattering problem. Each simulation showed the usefulness of the present scheme by yielding the steady scattering states in agreement with exact ones.