C.J Park

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

Abstract 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 Simpsons rule. From the analysis of spatial truncation error, the accuracy of the linear multiple balance scheme is O (Δ 4 ) whereas that of diamond differencing is O (Δ 2 ). 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 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.

Convergence Analysis of Additive Angular Dependent Rebalance Acceleration for the Discrete Ordinates Transport Calculations

Abstract In solving the discrete ordinates neutron transport equation, the additive angular dependent rebalance (AADR) acceleration method proposed by the authors previously is simple to implement, unconditionally stable, and very effective. For slab geometry problems, it is demonstrated via Fourier analysis that the spectral radii of the AADR acceleration in S4-like and DP1-like rebalances as well as DP0-like rebalance are less than that of diffusion synthetic acceleration (DSA). This AADR acceleration method is easily extendable to DPN-like and low-order SN-like rebalancing, and it does not require consistent discretizations between the high-order and low-order equations as does DSA. The continuous Fourier analysis is also performed for rectangular geometry. This Fourier analysis shows that the AADR with directional S2-like weighting functions, which uses two different rebalance factors for the x and y directions per octant, provides better results than the AADR with the normal S2-like weighting functions, which uses a single weighting function per octant. The low-order equation in AADR is solved by a preconditioned Bi-CGSTAB algorithm, which reduces computational burden significantly.