Extract of a 3D Cartesian grid. Unlike the general case presented in Fig. 1, we consider here the particular case of a medium with a double horizontal periodicity, in x-direction and in y-direction. In x-direction, one period is defined from x₀ to x_Nx. The plane x = x₀ is a boundary plane. The cell defined by indices (− 1,j,k), drawn with thick lines, is identical to the cell (N_x,j,k) (not drawn). The upwind specific intensity is known at M_u for the short-characteristic (M_u,M_c,M_d), since M_u belongs to a horizontal face. On the other hand, the upwind specific intensity is not known at for the short-characteristic , because belongs to a vertical boundary face. Therefore, the characteristic is prolonged up to the first intersection with a horizontal face, . See Sect. 7 for more explanations.