A Linear Algebraic Development of Diffusion Synthetic Acceleration for Three-Dimensional Transport Equations

Abstract

Linear algebraic formulations of discretized, mono-energetic, steady-state, linear Boltzmann transport equations (BTE) in three dimensions are presented. The discretizations consist of a discrete ordinates collocation in angle and a Petrov–Galerkin finite element method in space. A matrix development of diffusion synthetic acceleration (DSA) is given for three-dimensional (3-D) rectangular geometry. It is shown that the DSA “consistently” differenced diffusion approximation to the BTE is actually singular in three dimensions, although the DSA preconditioner itself is nonsingular. Numerical results are presented that demonstrate the effectiveness of the derived DSA preconditioner in the thick and thin limits for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux prescribed at the boundaries.