Solving nonlinear discretizations of [formula omitted] transport calculations

Abstract

In this paper, we investigate iterative strategies to converge transport solutions using nonlinear discretization schemes, specifically the adaptive-weighted diamond-difference method and a set-to-zero fixup approach for the linear discontinuous method compatible with arbitrary polytopal cells. The nonlinear nature of these schemes precludes the use of linear methods, such as highly effective Krylov iterative solvers like GMRES. Instead, nonlinear Newton-like solvers such as Broyden’s method, nonlinear Krylov acceleration, and the Jacobian-free Newton-Krylov method are analyzed in conjunction with physics-based diffusion acceleration schemes. The Richardson method (i.e., source iteration) is also analyzed for comparison. Numerical experiments (both monoenergetic and multigroup) demonstrate effective reduction in transport work by the use of appropriately chosen iterative schemes, including choosing sufficiently consistent diffusion acceleration. These results are analogous to linear transport operators, and they can be used as a point of reference in determining solution strategies for other nonlinear transport discretization schemes.