Numerical Solution Algorithms for a P_(N-1) -Equivalent S_N Angular Discretization of the Transport Equation in One-Dimensional Spherical Geometry, invited

摘要

We propose solution algorithms for solving an effective steady-state, linear transport equation for high energy density radiative transfer calculations in one-dimensional spherical geometry. We derive spherical geometry S_N equations that are P_(N-1)-equivalent with Gauss quadrature, using relationships given in to compute the angular derivatives in the P_(N-1)-equivalent S_N equations exactly. Traditional spherical geometry methods typically introduce a first order approximation in the angular redistribution term. Our exact treatment incurs the expense of having to express the angular derivatives in terms of all the angular fluxes at a spatial point. The resulting sweep operator is not lower-triangular and cannot be inverted efficiently. Splitting the angular redistribution operator to make the sweep operator lower-triangular requires that we write in terms of the angular fluxes rather than the scalar fluxes. We present and compare two splittings, one corresponding to Richardson iteration (source iteration) and the other to Gauss-Seidel iteration. We cast the algorithms as preconditioners to a Krylov iterative method, with the expectation that Krylov methods will be more efficient, and possibly more robust, than the classical iterations.