A Linear Algebraic Analysis of Diffusion Synthetic Acceleration for the Boltzmann Transport Equation II: The Simple Corner Balance Method

Abstract

In this paper we apply the development and linear algebraic analysis of the diffusion synthetic acceleration method presented in [S. F. Ashby, P. N. Brown, M. R. Dorr, and A. C. Hindmarsh, SIAM J. Numer. Anal., 32 (1995), pp. 128–178] to a different spatial discretization. Our model equation is the monoenergetic, steady-state, linear Boltzmann transport equation in slab geometry. The discretization consists of a discrete ordinates collocation in angle and the simple corner balance method in space. By expressing diffusion synthetic acceleration in this formalism, asymptotic results are obtained that prove the effectiveness of the associated preconditioner in various limiting cases, including the asymptotic diffusion limit. These results hold for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux at the boundaries. Numerical results confirm the theoretical estimates.