We investigate the question whether the D-1 approximation to the linear Boltzmann equation (LBE) can be used as an amendment to the diffusion approximation instead of the P-1 and P-1/3 solutions. We study two initial value problems in 3D: for a localised source of short duration and a localised initial distribution. We have found that at short times the D-1 solutions do not accurately approximate the solutions of the LBE, but the same is true for the diffusion approximation. However, at longer times the D-1 approximation is much better than the diffusion one. In contrast to the P-1 and P-1/3 solutions, which exhibit evident unphysical behaviour in the problems we have considered, the D-1 solutions manifest virtually "nonnegative" behaviour. Besides, the D-1 approximation is much better than the P-1 and P-1/3 solutions at all times. (C) 2015 Elsevier Ltd. All rights reserved.