Performance Evaluation of MPI/OpenMP Algorithm for 3D Time Dependent Problems

We consider the 3D time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, approached in terms of velocity and pressure. In a parallel algorithm, based on a novel direction splitting approach, the pressure equation is derived from a perturbed form of the continuity equation, in which the incompressibility constraint is penalized in a negative norm induced by the direction splitting. In order to achieve good parallel performance, the solution of the Poison problem for the pressure correction is replaced by solving a sequence of one-dimensional second order elliptic boundary value problems in each spatial direction. The parallel code was developed using MPI and OpenMP and tested on modern computer systems. The performed tests illustrate the parallel efficiency, and the scalability, of the direction-splitting based algorithm.

[1]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[2]  W. Walker,et al.  Mpi: a Standard Message Passing Interface 1 Mpi: a Standard Message Passing Interface , 1996 .

[3]  Maria Ganzha,et al.  Performance Analysis of Parallel Alternating Directions Algorithm for Time Dependent Problems , 2011, PPAM.

[4]  Barbara Chapman,et al.  Using OpenMP - portable shared memory parallel programming , 2007, Scientific and engineering computation.

[5]  Abner J. Salgado,et al.  A fractional step method based on a pressure Poisson equation for incompressible flows with variable density , 2008 .

[6]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[7]  Abner J. Salgado,et al.  A splitting method for incompressible flows with variable density based on a pressure Poisson equation , 2009, J. Comput. Phys..

[8]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) , 1969 .

[9]  Dan Nagle,et al.  MPI -- The Complete Reference, Vol. 1, The MPI Core, 2nd ed., Scientific and Engineering Computation Series, by Marc Snir, Steve Otto, Steven Huss-Lederman, David Walker and Jack Dongarra , 2005 .

[10]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[11]  Rohit Chandra,et al.  Parallel programming in openMP , 2000 .

[12]  Jean-Luc Guermond,et al.  A new class of fractional step techniques for the incompressible Navier–Stokes equations using direction splitting , 2010 .

[13]  Jean-Luc Guermond,et al.  A new class of massively parallel direction splitting for the incompressible Navier―Stokes equations , 2011 .

[14]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .