A full-approximation storage multigrid method for solving the steady-state 2-d incompressible Navier-Stokes equations on stag-solved problem.
gered grids has been implemented in Fortran on the CM-5,using
Parallel computing is advantageous because of the large the array aliasing feature in CM-Fortran to avoid declaring fine-gridproblem sizes which can be accommodated. The computasized arrays on all levels while still allowing a variable number tional speeds (e.g., Mflops) which can be obtained, howof grid levels. Thus, the storage cost scales with the number of ever, are strongly algorithm-dependent and, for a particuunknowns,allowing us to consider significantly larger problems than would otherwise be possible. Timings over a range of problem lar algorithm, strongly problem-size dependent. The sizes and numbers of processors, up to 4096 ฯซ 4096 on 512 nodes, effective computation rate depends on the relative show that the smoothing procedure, a pressure-correction techamounts and speeds of computation and interprocessor nique, is scalable and that the restriction and prolongation steps communication. Raw communication speeds are typically are nearly so. The performance obtained for the multigrid method orders of magnitude slower than floating-point operations.
is 333 Mflops out of the theoretical peak 4 Gflops on a 32-node CM-5.
In comparison, a single-grid computation obtained 420 Mflops. Thus more often than not the communication steps in the The decrease is due to the inefficiency of the smoothing iterations algorithm, and the network performance for these steps, on the coarse grid levels. W cycles cost much more and are much strongly influence the parallel run time.
less efficient than V cycles, due to the increased contribution from Because of the aforementioned advantages of multigrid the coarse grids. The convergence rate characteristics of the presmethods and parallel computing, there has been much sure-correction multigrid method are investigated in a Re ฯญ 5000 lid-driven cavity flow and a Re ฯญ 300 symmetric backward-facing interest in developing and testing parallel multigrid prostep flow, using either a defect-correction scheme or a second-order grams, as summarized in Refs. . A key issue, upwind scheme. A heuristic technique relating the convergence one which appears in more than one context in parallel tolerances for the coarse grids to the truncation error of the discreticomputing, is scalability. A scalable parallel architecture is zation has been found effective and robust. With second-order upone whose interprocessor communication network prowinding on all grid levels, a 5-level 320ฯซ 80 step flow solution was obtained in 20 V cycles, which corresponds to a smoothing rate of vides a fixed bandwidth for a particular communication 0.7, and required 25 s on a 32-node CM-5. Overall, the convergence operation, e.g., all-to-all broadcast, as the number of prorates obtained in the present work are comparable to the most cessors grows. Scalable parallel algorithms are, in an absocompetitive findings reported in the literature. แฎ 1996 Academic lute sense, those whose computational and communication Press, Inc.
complexity both depend only on the problem size per processor. All the processors must be involved for all the steps. In a relative sense, scalable algorithms are those which