A parallel multigrid solver for 3D convection and convection–diffusion problems

Abstract

Multigrid algorithms are known to be highly efficient in solving systems of elliptic equations. However, standard multigrid algorithms fail to achieve optimal grid-independent convergence rates in solving non-elliptic problems. In many practical cases, the non-elliptic part of a problem is represented by the convection operator. Downstream marching, when it is viable, is the simplest and most efficient way to solve this operator. However, in a parallel setting, the sequential nature of marching degrades the efficiency of the algorithm. The aim of this paper is to present, evaluate and analyze an alternative highly parallel multigrid method for 3D convection-dominated problems. This method employs semicoarsening, a four-color plane-implicit smoother, and discretization rules allowing the same cross-characteristic interactions on all the grids involved to be maintained. The load imbalance below the critical level is the main source of inefficiency in its parallel implementation. After comparing several strategies to relieve the load imbalance, a hybrid smoother that degrades the convergence properties of the method but improves its granularity has been found to be the best choice. The numerical and parallel properties of the multigrid algorithm with the four-color and hybrid smoothers are studied on SGI Origin 2000 and Cray T3E systems.