A linear multiple balance method with high order accuracy for discrete ordinates neutron transport equations

A linear multiple balance method (LMB) is developed to provide more accurate and positive solutions for the discrete ordinates neutron transport equations. In this multiple balance approach, one mesh cell is divided into two subcells with quadratic approximation for the angular flux distribution. Four multiple balance equations are used to relate center angular flux with average angular flux by Simpsonʹs rule. From the analysis of spatial truncation error, the accuracy of the linear multiple balance scheme is O(Δ4) whereas that of diamond differencing is O(Δ2). The positivity of the method is also stronger than that of diamond differencing. To accelerate the linear multiple balance method, we also describe an additive angular dependent rebalance factor scheme which combines a modified boundary projection acceleration scheme and the angular dependent rebalance factor acceleration scheme. It is demonstrated, via Fourier analysis of a simple model problem as well as numerical calculations, that the additive angular dependent rebalance factor acceleration scheme is unconditionally stable with spectral radius <0.2069 c (c being the scattering ratio). The numerical results tested so far on slab-geometry discrete ordinates transport problems show that the solution method of linear multiple balance with additive angular rebalance acceleration is effective and sufficiently efficient.