Acceleration schemes for the discrete ordinates method

Discrete ordinates (DO) methods have been developed to solve the multidimensional radiative transport equation (RTE) for applications including combustion processes and other combined-mode heat transfer cases. The convergence of DO methods is known to degrade for optical thicknesses greater than unity, which occur for example in a flame. Acceleration schemes have been developed for use in neutron transport applications, but little work has been done to accelerate convergence of the RTE for radiative heat transfer applications. This article presents several acceleration schemes for the RTE, including successive overrelaxation, synthetic acceleration, and mesh rebalance methods. Solution convergence is discussed and demonstrated using two- and three-dimensional examples. Although all methods improve convergence, the mesh rebalance method improves the RTE convergence best. For some conditions, the rebalance method improves convergence dramatically, reducing RTE iterations by an order of magnitude. However, the mesh rebalance method fails to produce convergence of the RTE for large optical thicknesses and fine mesh discretizations. Examples are used to demonstrate that improved convergence can be obtained by solving the rebalance equation on a coarser grid, which Is determined by regrouping the base RTE grid, until an optical thickness of near unity is obtained on the coarse grid, Based on these findings, a general solution strategy is discussed.