A Multigrid Framework for $S_n$ Discretizations of the Boltzmann Transport Equation

Abstract

The space-angle multiple/semicoarsening multigrid methods of [B. Lee, SIAM J. Sci. Comput., 31 (2010), pp. 4744--4773] for $S_n$ discretizations of the Boltzmann transport equation were developed assuming rediscretizations of the streaming-collision operators on the coarser spatial grids. These rediscretized operators are directly used in the kernels of the spatial coarse-level scalar flux integral equations. Since the computational procedure of these methods is essentially performed on these integral equations, it appears that the multigrid procedure is applied directly to the finest level scalar flux integral equation rather than to the finest level $S_n$ system. However, this procedure is actually implicitly applied to the finest level $S_n$ system. It would be beneficial to relate the multiple/semicoarsening methods directly to the full $S_n$ system or, better yet, to develop a general multigrid framework for the full $S_n$ system, of which the multiple/semicoarsening methods are particular cases. This general framework can help guide the development of multigrid algorithms for more general $S_n$ discretization settings, such as when spatial rediscretization of the integral equation kernel cannot be applied. In this latter case, applying an operator-dependent Galerkin coarsening procedure to the integral operator is difficult because the kernel is dense and often given only by its action, i.e., numerically sweeping over all the streaming-collision operators. In this paper, we develop and analyze a multigrid framework that is applied directly to the $S_n$ system and encompasses the multiple/semicoarsening methods. We further relate this framework to a multigrid method that directly coarsens the integral operator using a Galerkin coarsening procedure. A key component in developing this framework is to relate the smoothing of the multiple/semicoarsening methods to an approximate Schur complement smoothing on the full $S_n$ system. In this paper, this relation is established, and the smoothing and approximation properties are derived for the multigrid procedure induced by this framework.