An ADI-Like Preconditioner for Boltzmann Transport

Abstract

Numerical modeling of radiation transport and fluid dynamics are central to many fields of science. This paper studies the application of preconditioned iterative methods to time-dependent coupled hydrodynamics and radiation (Boltzmann) transport. In particular, we focus on the solution of Jacobian matrices in full Newton iterations in the modeling of neutrino (radiation) transport in core collapse supernovae. This is a demanding radiation hydrodynamics application, requiring accurate time-dependent neutrino transport coupled to hydrodynamics driven by this transport. Given a fully implicit finite differencing of the transport equations, the Jacobian matrices have a special block tridiagonal structure. The off-diagonal blocks are exactly diagonal and correspond to the spatial transport of neutrinos and the coupling of adjacent spatial zones owing to this transport. The dense diagonal blocks, on the other hand, arise from the coupling between the neutrino direction cosines and energy, as a result of neutrino emission, absorption, and scattering, in the multiple species multiple energy group discrete ordinates formulation for solving the Boltzmann transport equations. We consider an effective alternating direct implicit (ADI)-like preconditioner where we alternately sweep along the one-dimensional spatial direction and separately handle the coupling among multidirection multigroup variables within each spatial discretization cell. Numerical examples are given to illustrate the fast convergence and efficient parallel implementation. The results suggest that even simple fixed point iteration can achieve a similar convergence rate as {\tt GMRES} and {\tt BICGSTAB}.