Nonlinear and Linear α-Weighted Methods for Particle Transport Problems

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

“…The relative pointwise convergence criterion with˜ = 10 −12 is imposed. Table 3.3 shows the numbers of iterations for the proposed NWF method, as well as for the α-WN method with α=0.366, the QD and DSA methods that were taken from [12,17]. The results of problems 1 and 2 show that the developed method converges fast, and its convergence properties are close to those of the α-WN method with α=0.366, the QD and DSA methods.…”

“…We also show the results calculated by means of other flux methods, namely, the FF, SF, and α-WN (α=0.366) methods. The low-order equations of these methods were discretized by the lumped LD method, and the SC method was used for the transport equation [12]. The relative pointwise convergence criterion with = 10 −7 is used.…”

“…To accelerate transport iterations, nonlinear projective-iteration methods have been developed [8,5]. Among these methods are the quasidiffusion method [4], the first-flux (FF) (averaged flux) method [9,10], second-flux (SF) method [11], α-weighted nonlinear (α-WN) methods [12], nonlinear S 2 -like methods [13,14] and others [5]. The low-order equations of the flux methods are formulated for the partial scalar fluxes.…”

“…For stability of these methods, there is no need to discretize the transport and low-order equations consistently. However, neither the FF method nor the SF method satisfies the diffusion limit unless the discretization of the low-order equations is consistent with the transport differencing scheme, which itself satisfies the diffusion limit [8,12].…”

“…The family of the nonlinear α-weighted methods is the generalization of the flux methods [12]. In case of 1D slab geometry, these methods are derived by integrating the transport equation over −1 ≤ µ ≤ 0 and 0 ≤ µ ≤ 1 with weight |µ| α , α ≥ 0.…”