A LINEAR ALGEBRAIC DEVELOPMENT OF DIFFUSION SYNTHETIC ACCELERATION FOR 3-DIMENSIONAL TRANSPORT-EQUATIONS

Linear algebraic formulations of discretized, mono-energetic, steady-s tate, linear Boltzmann transport equations (BTE) in three dimensions a re presented. The discretizations consist of a discrete ordinates coll ocation in angle and a Petrov-Galerkin finite element method in space. A matrix development of diffusion synthetic acceleration (DSA) is giv en for three-dimensional (3-D) rectangular geometry. It is shown that the DSA ''consistently'' differenced diffusion approximation to the BT E is actually singular in three dimensions, although the DSA precondit ioner itself is nonsingular. Numerical results are presented that demo nstrate the effectiveness of the derived DSA preconditioner in the thi ck and thin limits for problems with nonconstant coefficients and nonu niform spatial zoning posed on finite domains with an incident flux pr escribed at the boundaries.