The implementation of a hybrid spectral/finite-element discretization on the unsteady, incompressible, Navier-Stokes equations with a semi-implicit time-stepping method, an explicit treatment of the advective terms, and an implicit treatment of the pressure and viscous terms leads to an algorithm capable of calculating 3D flows over complex 2D geometries. This also results in multiple Fourier mode linear systems which must be solved at every timestep, which naturally leads to two parallelization approaches: Fourier space partitioning, where each processor individually and simultaneously solves a linear system, and physical space partitioning, where all processors collectively solve each linear system, sequentially advancing through Fourier modes. These two parallelization approaches are compared based upon computational cost using multiple solvers: direct sparse LU, smoothed aggregation AMG, and single-level ILUT preconditioned GMRES; and on two supercomputers of different memory architecture(distributed and shared memory). This study revealed Fourier space partitioning outperforms physical space partitioning in all problems analyzed, and scales more efficiently as well. These differences were more dramatic on the distributed memory platform than the shared memory platform. Another study compares the previously mentioned solvers along with one additional solver, pointwise AMG, in Fourier space partitioning without parallelization to better understand computational scaling for problems with large meshes. It was found that the direct sparse LU solver performed well in terms of computational time, scaled linearly, but had very high memory usage which scaled in a super-linear manner. The single-level ILUT preconditioned GMRES solver required the least amount of memory, which also scaled linearly, but only had acceptable performance in terms of computational time for coarse meshes. Both AMG methods scaled linearly in both memory usage and time, and were comparable to the direct sparse LU solver in terms of computational time. The results of these studies are particularly useful for implementation of this algorithm on challenging and complex flows, especially direct numerical and large-eddy simulations. Reducing computational cost allows the analysis and understanding of more flows of practical interest.



College and Department

Ira A. Fulton College of Engineering and Technology; Mechanical Engineering



Date Submitted


Document Type





algebraic multigrid, Navier-Stokes equations, large eddy simulation, direct numerical simulation