The Effectiveness of P₂ and Simplified P₂ Synthetic Accelerations in the Solutions of Discrete Ordinates Transport Equations

Using the operator form of a synthetic acceleration, the P₁ acceleration [diffusion synthetic acceleration (DSA)] and P₂ acceleration schemes for one-dimensional slab and the P₁ and simplified P₂ acceleration schemes for two-dimensional x-y geometry are derived. The convergence rate of each scheme for a simple model problem is compared, and the result is generalized by performing a Fourier analysis. In the one-dimensional case, the new second-moment P₂ acceleration outperforms an earlier third-moment P₂ acceleration developed by Miller and Larsen. However, it is still less efficient than P₁ acceleration. Similar results show that the P₁ acceleration converges faster than the simplified P₂ acceleration in two-dimensional x-y geometry. These results confirm that one cannot simply assume that replacement of the DSA method with a higher order operator will lead to a smaller spectral radius.