Red-Black Response Matrix Acceleration by Transformation of Interface Variables

Red-black algorithms for solving response matrix equations in one- and two-dimensional diffusion theory are examined. The definition of the partial currents in terms of the scalar flux and net currents is altered to introduce an acceleration parameter that modifies the values of the response matrix elements while leaving the flux and net current solutions unchanged. The acceleration parameter is selected for response matrices derived analytically for slab geometry and from the variational nodal method for both slab and x-y geometries to minimize the spectral radius of the red-black iteration matrix for homogeneous media. The optimal value is shown to be independent of the mesh spacing in the fine mesh limit and to be a function only of c, the scattering-to-total cross section ratio. The method is then generalized to treat multiregion problems by formulating an approximate expression for the optimum acceleration parameter and demonstrated for a series of benchmark diffusion problems.