AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport

Abstract

We develop a reduction multigrid based on approximate ideal restriction (AIR) for use with asymmetric linear systems. We use fixed-order GMRES polynomials to approximate $A_\textrm{ff}^{-1}$ and we use these polynomials to build grid transfer operators and perform F-point smoothing. We can also apply a fixed sparsity to these polynomials to prevent fill-in. When applied in the streaming limit of the Boltzmann Transport Equation (BTE), with a P$^0$ angular discretisation and a low-memory spatial discretisation on unstructured grids, this "AIRG" multigrid used as a preconditioner to an outer GMRES iteration outperforms the lAIR implementation in hypre, with two to three times less work. AIRG is very close to scalable; we find either fixed work in the solve with slight growth in the setup, or slight growth in the solve with fixed work in the setup when using fixed sparsity. Using fixed sparsity we see less than 20% growth in the work of the solve with either 6 levels of spatial refinement or 3 levels of angular refinement. In problems with scattering AIRG performs as well as lAIR, but using the full matrix with scattering is not scalable. We then present an iterative method designed for use with scattering which uses the additive combination of two fixed-sparsity preconditioners applied to the angular flux; a single AIRG V-cycle on the streaming/removal operator and a DSA method with a CG FEM. We find with space or angle refinement our iterative method is very close to scalable with fixed memory use.