Performance of Parallel Approximate Ideal Restriction Multigrid for Transport Applications

Abstract

Algebraic multigrid (AMG) methods have been widely used to solve systems arising from the discretization of elliptic partial differential equations. In serial, AMG algorithms scale linearly with problem size. In parallel, communication costs scale logarithmically with the number of processors. Recently, a classical AMG method based on approximate ideal restriction (AIR) was developed for nonsymmetric matrices. AIR has already been shown to be effective for solving the linear systems arising from upwind discontinuous Galerkin (DG) finite element discretization of advection-diffusion problems, including the hyperbolic limit of pure advection. A new parallel version of AIR, pAIR, has been implemented in the hypre library. In this thesis, pAIR is tested for use solving the source iteration equations of the SN approximations to the transport equation. The performance is investigated with various meshes in two and three dimensions. Detailed profiling of parallel performance is also conducted to identify the most important areas for algorithm improvements. An improvement to the Local Ideal Approximate Restriction algorithm is introduced and discussed. Weak scaling results to 4,096 processors are presented. These results show total solve growing logarithmically with the number of processors used. Importantly, this result is shown on both uniform grids and unstructured grids in three dimensions. The unstructured mesh did not include reentrant cells.