Probing the D1 approximation to the linear Boltzmann equation in 3D

We investigate the question whether the D1 approximation to the linear Boltzmann equation (LBE) can be used as an amendment to the diffusion approximation instead of the P1 and P1/3 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 D1 solutions do not accurately approximate the solutions of the LBE, but the same is true for the diffusion approximation. However, at longer times the D1 approximation is much better than the diffusion one. In contrast to the P1 and P1/3 solutions, which exhibit evident unphysical behaviour in the problems we have considered, the D1 solutions manifest virtually “nonnegative” behaviour. Besides, the D1 approximation is much better than the P1 and P1/3 solutions at all times.