The Quasi-Diffusion method for solving transport problems in planar and spherical geometries

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Abstract
The Quasi-Diffusion (QD) method [1]–[3] is a nonlinear algorithm for solving linear transport problems. Because the QD method utilizes both a transport sweep and a diffusion calculation within each iteration, it is operationally more complex than the Source Iteration (SI) method, which utilizes only a transport sweep within each iteration. However, the QD method often converges rapidly and with high accuracy, especially for optically thick regions with scattering ratios c close to unity; these are the regions for which acceleration is most needed. A difficulty with the QD method is that because it is nonlinear, every scalar flux iterate must be positive at each point in the system. Also, the formulation of diffusion boundary conditions to optimize accuracy and speed of convergence is not obvious. In this paper, we consider both of these issues. Specifically, we propose a new formulation of the QD method in spherical geometry to guarantee positivity of the analytic solution, and we propose new diffusion boundary conditions in planar and spherical geometry that lead to more accurate and efficient solution algorithms. Also, we discuss ways to accurately and positively discretize the transport and diffusion equations, and we give extensive numerical results.