Searchlight-beam test for direction n = (1, 0, 1) and a typical grid with N_x = N_y = N_z = 133 grid points. Upper panel: contour plot of the specific intensity as calculated with the SC method using Bézier interpolations in the xz-plane (cf. Paper I, Fig. 3, for the finite-volume method). Middle panel: specific intensity through the perpendicular area indicated by the straight line in the upper panel. The blue, red, and green profiles correspond to the FVM, SClin, and SCbez methods, respectively. The dashed line indicates the theoretical profile. Bottom panel: as middle panel, but along the centre of the searchlight beam. We note that the SC methods reproduce the exact solution at the centre of the beam, whereas the FVM solution decreases significantly for r ≳ 2.5 R_*.