A Parallel Multigrid Finite-Volume Solver on a Collocated Grid for Incompressible Navier-Stokes Equations

Abstract

Multigrid techniques are widely used to accelerate the convergence of iterative solvers. Serial multigrid solvers have been efficiently applied to a broad class of problems, including fluid flows governed by incompressible Navier-Stokes equations. With the recent advances in high-performance computing (HPC), there is an ever-increasing need for using multiple processors to solve computationally demanding problems. Thus, it is imperative that new algorithms be developed to run the multigrid solvers on parallel machines. In this work, we have developed a parallel finite-volume multigrid solver to simulate incompressible viscous flows in a collocated grid. The coarse-grid equations are derived from a pressure-based algorithm (SIMPLE). A domain decomposition technique is applied to parallelize the solver using a Message Passing Interface (MPI) library. The multigrid performance of the parallel solver has been tested on a lid-driven cavity flow. The scalability of the parallel code on both single- and multigrid solvers was tested and the characteristics were analyzed. A high-fidelity benchmark solution for lid-driven cavity flow problem in a 1,024 × 1,024 grid is presented for a range of Reynolds numbers. Parallel multigrid speedup as high as three orders of magnitude is achieved for low-Reynolds-number flows. The optimal multigrid efficiency is validated, i.e., the computational cost is shown to increase proportionally with the problem size.

Acknowledgments

The authors are grateful to Professor Arun Srinivasa for his insightful lecture on multigrid techniques. The Texas A&M Supercomputing Facility (http://sc.tamu.edu) is gratefully acknowledged for providing computing resources useful in conducting the research reported in this article.

Notes

¹ MPI_Allreduce combines data from all processes and distributes the reduced data back to all processes [40].

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