Coarsening and parallelism with reduction multigrids for hyperbolic Boltzmann transport

Abstract

Reduction multigrids have recently shown good performance in hyperbolic problems without the need for Gauss-Seidel smoothers. When applied to the hyperbolic limit of the Boltzmann Transport Equation (BTE), these methods result in very close to $\mathcal{O}(n)$ growth in work with problem size on unstructured grids. This scalability relies on the CF splitting producing an $A_\textrm{ff}$ block that is easy to invert. We introduce a parallel two-pass CF splitting designed to give diagonally dominant $A_\textrm{ff}$. The first pass computes a maximal independent set in the symmetrized strong connections. The second pass converts F-points to C-points based on the row-wise diagonal dominance of $A_\textrm{ff}$. We find this two-pass CF splitting outperforms common CF splittings available in hypre. Furthermore, parallelisation of reduction multigrids in hyperbolic problems is difficult as we require both long-range grid-transfer operators and slow coarsenings (with rates of $\sim$1/2 in both 2D and 3D). We find that good parallel performance in the setup and solve is dependent on several factors: repartitioning the coarse grids, reducing the number of active MPI ranks as we coarsen, truncating the multigrid hierarchy and applying a GMRES polynomial as a coarse-grid solver. We compare the performance of two different reduction multigrids, AIRG (that we developed previously) and the hypre implementation of $\ell$AIR. In the streaming limit with AIRG, we demonstrate 81\% weak scaling efficiency in the solve from 2 to 64 nodes (256 to 8196 cores) with only 8.8k unknowns per core, with solve times up to 5.9$\times$ smaller than the $\ell$AIR implementation in hypre.