Nonlinear and linear-weighted methods for particle transport problems

A parametrized family of iterative methods for the planar-geometry transport equation is proposed. This family is a generalization of previously proposed nonlinear flux methods. The new methods are derived by integrating the 1D transport equation over −1≤μ≤0 and 0≤μ≤1 with weight |μ|α, α≥0. Both nonlinear and linear methods are developed. The convergence properties of the proposed methods are studied theoretically by means of a Fourier stability analysis. The optimum value of α that provides the best convergence rate is derived. We also show that the convergence rates of nonlinear and linear methods are almost the same. Numerical results are presented to confirm these theoretical predictions.