@article{roberts_anistratov_2010, title={Nonlinear Weighted Flux Methods for Particle Transport Problems in Two-Dimensional Cartesian Geometry}, volume={165}, ISSN={["0029-5639"]}, DOI={10.13182/nse08-48}, abstractNote={Abstract A family of nonlinear weighted flux (NWF) methods for solving the transport equation in two-dimensional (2-D) Cartesian geometry is considered. The low-order equations of these methods are defined by means of special linear-fractional factors that are determined by the high-order transport solution. An asymptotic diffusion limit analysis is performed on methods with a general weight function. The analysis revealed conditions on the weight necessary for an accurate approximation of the diffusion equation in this limit. We study methods with weights defined by linear and bilinear functions of directional cosines. As a result, we developed 2-D NWF methods formulated with the low-order equations that give rise to the diffusion equation in optically thick diffusive regions if their factors are calculated by means of the leading-order transport solution. The inherent asymptotic boundary conditions for the NWF methods are analyzed. Numerical results are presented to confirm theoretical results and demonstrate performance of the proposed methods.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Roberts, Loren and Anistratov, Dmitriy Y.}, year={2010}, month={Jun}, pages={133–148} }
 @article{roberts_anistratov_2007, title={Nonlinear weighted flux methods for particle transport problems}, volume={36}, ISSN={["1532-2424"]}, DOI={10.1080/00411450701703647}, abstractNote={A new parameterized family of iterative methods for the 1‐D slab geometry transport equation is proposed. The new methods are derived by integrating the transport equation over −1≤μ≤0 and 0≤μ≤1 with weight 1+β|μ|α, where α≥0. The asymptotic diffusion analysis enables us to determine a particular method of this family the solution of which satisfies a good approximation of both the diffusion equation and asymptotic boundary condition in the diffusive regions. Note that none of the α‐weighted nonlinear methods possesses this combination of properties. The convergence properties of the proposed method are similar to the properties of the diffusion‐synthetic acceleration (DSA), quasi‐diffusion, and DSA‐like α‐weighted nonlinear methods. Numerical results are presented to demonstrate the performance of the derived method.}, number={7}, journal={TRANSPORT THEORY AND STATISTICAL PHYSICS}, author={Roberts, L. and Anistratov, D. Y.}, year={2007}, pages={589–608} }