Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Abstract

Using discontinuous Galerkin finite-element methods to solve the hyperbolic components of the transport operator in the 1-D spherically symmetric case, the convergence behavior of various iterative methods are compared and evaluated. Diffusion synthetic accelerator, a preconditioner based on the diffusion approximation to the transport equation, is presented formally for this geometry for the first time. Compared with classical, finite-difference like methods (diamond difference methods), it is found that DG diffusion based preconditioners performed extremely well in resolving problems with strong scattering effects and material discontinuities.Keywords Neutron transport equation Á Iterative methods Á Integro-differential operator Discontinuous Galerkin Methods with Isotropic ScatteringThe Boltzmann operator considered here iswhere r t C r s C 0. The notation S½w r s ðrÞ Z þ1 À1 wðr;lÞ dl 2 ¼ r s ðrÞuðrÞ ð 2Þ may be used on occasion where the scalar flux is defined as uðrÞ ¼ Z þ1 À1 wðr;lÞ dl 2 :The streaming-collision operator T ¼ B À S can be written as T ¼ j Á r þ r t where j ¼ ðl; 1Àl 2 r Þ: The Boltzmann transport equation (BTE) problem properly stated is as follows: Given a domain D ¼ ½0; 1  ½À1; 1 in (r, l)-space, and source term q(r), find a solution w(r, l) such thatNote that in many of the examples it will be commonly assumed that r t , q, and r s are all isotropic (independent of l), but in all cases these quantities are non-negative. Brief Overview of Previous EffortsIn 1973, Lesaint and Ravaint [12] produced one of the seminal studies of the convergence properties of the discontinuous Galerkin (DG) method as applied to the nonscattering constant coefficient 2-D Cartesian transport equation