Efficient Parallel Multigrid Solver for 3-D Convection-dominated Problems

摘要

Multigrid algorithms are known to be highly efficient in solving systems of elliptic equations. However, standard multi grid 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 report is to present, evaluate and analyze an alternative highly parallel multi grid method for 3-D convection-dominated problems. This method employs semi coarsening, a four-color plane-implicit smoother, and discretization rules allowing the same cross-characteristic interactions on all the grids involved to be maintained. The resulting multigrid solver exhibits a fast grid-independent convergence rate for solving the convection-diffusion operator on cell-centered grids with stretching. The load imbalance below the critical level is the main source of inefficiency in its parallel implementation. A hybrid smoother that degrades the convergence properties of the method but improves its granularity has been found to be the best choice in a parallel setting. The numerical and parallel properties of the multi grid algorithm with the four-color and hybrid smoothers are studied on SGI Origin 2000 and Cray T3E systems.