Diffusion Synthetic Acceleration Methods for the Diamond-Differenced Discrete-Ordinates Equations

We investigate a class of acceleration schemes that resemble the conventional synthetic method in that they utilize the diffusion operator in the transport iteration schemes. These schemes are not dependent on diffusion theory as being a good approximation to transport theory; they only make use of the diffusion equation form. The accelerated iteration involves alternate diffusion and transport solutions where coupling between the equations is achieved using a correction term applied to either (a) the diffusion coefficient, (b) the removal cross section, or (c) the source of the diffusion equation. The methods involving the modification of the diffusion coefficient and of the removal term yield nonlinear acceleration schemes and are used in k_eff calculations, while the source term modification approach is linear at least before discretization and is used for inhomogeneous source problems. A careful analysis shows that there is a preferred differencing method that eliminates the previously observed instability of the conventional synthetic method. Using this preferred difference scheme results in an acceleration method that is at the same time stable and efficient. This preferred difference approach renders the source correction scheme, which is linear in its continuous form and nonlinear in its differenced form. An additional feature of these approaches is that they can be used as schemes for obtaining improved diffusion solutions for approximately twice the cost of a diffusion calculation. Numerical experimentation on a wide range of problems in one and two dimensions indicates that improvement from a factor of from 2 to 10 over rebalance or Chebyshev acceleration is obtained. The improvement is most pronounced in problems with large regions of scattering material where the unaccelerated transport solutions converge very slowly.