Parallel alternating directions algorithm for 3D Stokes equation

We consider the 3D time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. For this problem, a parallel algorithm, based on a recently proposed direction splitting approach, is applied. Here, the pressure equation is derived from a perturbed form of the continuity equation, where the incompressibility constraint is penalized in a negative norm induced by the direction splitting. The scheme used in the algorithm is composed of: (a) pressure prediction, (b) velocity update, (c) penalty step, and (d) pressure correction. In order to achieve good parallel performance, the solution of the Poison problem for the pressure correction is replaced by a solution to a sequence of one-dimensional second order elliptic boundary value problems (in each spatial direction). The efficiency and scalability of the proposed approach are tested on two distinct parallel computers and the experimental results are analyzed.

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

[2]  P. G. Ciarlet,et al.  Numerical Methods for Fluids, Part 3 , 2003 .

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

[4]  Jaime Peraire,et al.  An implicit finite element method for high speed flows , 1991 .

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  P. G. Ciarlet,et al.  Numerical methods for fluids , 2003 .

[7]  C. Vuik,et al.  Preconditioners for the Steady Incompressible Navier-Stokes Problem , 2008 .

[8]  Vít Dolejší,et al.  On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow , 2002 .

[9]  Jack Dongarra,et al.  LAPACK Users' Guide, 3rd ed. , 1999 .

[10]  Raphaèle Herbin,et al.  Mathematical Modelling and Numerical Analysis Modélisation Mathématique et Analyse Numérique Will be set by the publisher ON A STABILIZED COLOCATED FINITE VOLUME SCHEME FOR THE , 2013 .

[11]  S. Turek.  Efficient solvers for incompressible flow problems: An algorithmic approach . . , 1998 .

[12]  Jie Shen,et al.  On the error estimates for the rotational pressure-correction projection methods , 2003, Math. Comput..

[13]  C. Vuik,et al.  SIMPLE‐type preconditioners for the Oseen problem , 2009 .

[14]  R. Hartmann,et al.  Symmetric Interior Penalty DG Methods for the CompressibleNavier-Stokes Equations I: Method Formulation , 2005 .

[15]  N T Frink,et al.  Recent Progress Toward a Three-Dimensional Unstructured Navier-Stokes Flow Solver , 1994 .

[16]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

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

[18]  John N. Shadid,et al.  Least Squares Preconditioners for Stabilized Discretizations of the Navier-Stokes Equations , 2007, SIAM J. Sci. Comput..

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

[20]  J. Flaherty,et al.  Parallel, adaptive finite element methods for conservation laws , 1994 .

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

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

[23]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[24]  Andrew J. Wathen,et al.  A Preconditioner for the Steady-State Navier-Stokes Equations , 2002, SIAM J. Sci. Comput..

[25]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[26]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[27]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[28]  Maxim A. Olshanskii,et al.  An Augmented Lagrangian-Based Approach to the Oseen Problem , 2006, SIAM J. Sci. Comput..

[29]  D. Spalding,et al.  A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows , 1972 .

[30]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[31]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[32]  J. Steger,et al.  Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods , 1981 .

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

[34]  Maria Ganzha,et al.  Parallel Performance Evaluation of MIC(0) Preconditioning Algorithm for Voxel µFE Simulation , 2009, PPAM.

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

[36]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .