A simple and stable sub-grid scheme is derived from the space-time finite integration method, where staircase-shaped edges can be used to construct a rectangle-based grid. Comparison with classical sub-grid methods shows that the proposed method suppresses unphysical wave reflection without interpolation of field values for the sub-grid connection. No numerical instability was observed even after 106 time steps. The proposed method was applied to the computation of a photonic band gap.