Improving Low Order, Linear, Positive Spatial Quadratures for the Partial Current Neutron Transport Method

AFIT researchers have developed a new approach to solving Discrete Ordinates equations, which approximate the linear Boltzmann Transport Equation (BTE). The usual approach is von Neumann iteration on the scattering source, which requires repeated sweeps through the spatial-angular grid. Acceptable convergence requires complicated and expensive acceleration schemes. The new approach, Partial-Current Transport (PCT) with Adaptive Distribution Iteration, eliminates scattering source iteration through matrix inversions and a reduced-size global linear algebra problem. It creates the needed matrices directly from the standard spatial quadratures used in the sweeping. Positivity, linearity, and (higher-than-first-order) accuracy are the key desirable qualities with all Discrete Ordinates methods, but all three, according to Lathrop [8], cannot be achieved simultaneously. If a high order accurate, linear method is used, it can produce negative fluxes. Non-linear methods have been developed that are high-order accurate and positive, but these methods are not widely accepted because the BTE is itself a linear equation. Positive, linear methods are available, but are only first-order accurate. The latter can achieve needed accuracy by using optically-thin cells, but with Source Iteration (SI), this requires a fine grid of many cells, hence large computational expense. My new approach is to partition an optically thick cell into 2N identical sub-cells. Each sub-cell is optically thin enough that first-order accurate spatial quadrature methods are sufficiently accurate as well as being linear and positive. The needed matrices are computed as before for a (thinnest) sub-cell. My algorithm combines the matrices for a pair of sub-cells to get the matrices for a single (merged) sub-cell twice as thick. Merging N times yields the matrices for the original cell. This allows PCT to solve the discrete ordinates equations with linearity, positivity, and sufficient accuracy without the high computational cost of increasing the number of cells by a factor of 2N