Application of the parallel diagonal dominant algorithm for the incompressible Navier-Stokes equations

Abstract The accuracy and the applicability of the parallel diagonal dominant (PDD) algorithm are explored for highly scalable computation of the incompressible Navier-Stokes equations which are integrated using a fully-implicit fractional-step method in parallel computational environments. The PDD algorithm is known to be applicable only for an evenly diagonal dominant matrix. In the present study, however, it is shown mathematically that the PDD algorithm is utilizable even for non-diagonal dominant matrices derived from discretization of incompressible momentum equations. The order of accuracy and the error characteristics are investigated in detail in terms of the Courant-Friedrichs-Lewy (CFL) number and the grid spacing by conducting simulations of decaying vortices in both two and three dimensions, flow in a lid-driven cavity, and flow over a circular cylinder. In order to reduce communication cost, which is one of bottlenecks in parallel computation, an aggregative data communication method is combined with the PDD algorithm. Parallel performance of the present PDD-based method is investigated by measuring the speedup, efficiency, overhead, and serial fraction.

[1]  Henk A. van der Vorst,et al.  Data transport in Wang's partition method , 1988, Parallel Comput..

[2]  Haecheon Choi,et al.  Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160 , 1998 .

[3]  J. Luo,et al.  Parallel Direct Method of DNS for Two-Dimensional Turbulent Rayleigh-Bénard Convection , 2018 .

[4]  Vipin Kumar,et al.  Isoefficiency: measuring the scalability of parallel algorithms and architectures , 1993, IEEE Parallel & Distributed Technology: Systems & Applications.

[5]  P. Moin,et al.  Effects of the Computational Time Step on Numerical Solutions of Turbulent Flow , 1994 .

[6]  B. Zeghmati,et al.  An efficient parallel high-order compact scheme for the 3D incompressible Navier–Stokes equations , 2017 .

[7]  Donghyun You,et al.  A multi-block ADI finite-volume method for incompressible Navier-Stokes equations in complex geometries , 2011, J. Comput. Phys..

[8]  John L. Gustafson,et al.  Reevaluating Amdahl's law , 1988, CACM.

[9]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[10]  Donghyun You,et al.  An implicit ghost-cell immersed boundary method for simulations of moving body problems with control of spurious force oscillations , 2013, J. Comput. Phys..

[11]  C. Williamson.  Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers , 1989, Journal of Fluid Mechanics.

[12]  Donghyun You,et al.  Control of Flow-Induced Noise Behind a Circular Cylinder Using Splitter Plates , 1998 .

[13]  Dimitri J. Mavriplis,et al.  Multigrid solution of the Navier-Stokes equations on triangular meshes , 1989 .

[14]  V. E. Henson,et al.  BoomerAMG: a parallel algebraic multigrid solver and preconditioner , 2002 .

[15]  S. Johnsson.  Solving tridiagonal systems on ensemble architectures , 1987 .

[16]  Eric F. van de Velde,et al.  Alternating-Direction Line-Relaxation Methods on Multicomputers , 1996, SIAM J. Sci. Comput..

[17]  Timothy J. Williams,et al.  Algorithm for Solving Tridiagonal Matrix Problems in Parallel , 1995, Parallel Comput..

[18]  H. H. Wang,et al.  A Parallel Method for Tridiagonal Equations , 1981, TOMS.

[19]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[20]  Lionel M. Ni,et al.  Parallel algorithms for solution of tridiagonal systems on multicomputers , 1989, ICS '89.

[21]  Petros Koumoutsakos,et al.  A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical flows at high Reynolds numbers , 2011, J. Comput. Phys..

[22]  Alan H. Karp,et al.  Measuring parallel processor performance , 1990, CACM.

[23]  Xian-He Sun.  Application and Accuracy of the Parallel Diagonal Dominant Algorithm , 1995, Parallel Comput..

[24]  J. Ferziger,et al.  An adaptive multigrid technique for the incompressible Navier-Stokes equations , 1989 .

[25]  J. Sherman,et al.  Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix , 1950 .

[27]  Akiyoshi Wakatani.  A parallel and scalable algorithm for ADI method with pre-propagation and message vectorization , 2004, Parallel Comput..

[28]  Lionel M. Ni,et al.  Efficient Tridiagonal Solvers on Multicomputers , 1992, IEEE Trans. Computers.