A parallel block-preconditioned GCR method for incompressible flow problems

Abstract Efficient parallel algorithms are required to simulate incompressible turbulent flows in complex two- and three-dimensional domains. The incompressible Navier–Stokes equations are discretized in general coordinates on a structured grid. For a flow on a general domain we use an unstructured decomposition of the domain into subdomains of simple shape, with a structured grid inside each subdomain. We have developed a parallel block-preconditioned GCR method to solve the resulting systems of linear equations. The method can be smoothly varied between a coarse grain parallel method in which the subdomain problems are solved accurately with an inner iteration process and a fine grain parallel method when only a preconditioner is used to approximate the solution on the blocks. Parallel performance results for Boussinesq flow in a cavity are included.

[1]  Cornelis W. Oosterlee,et al.  Invariant discretization of the incompressible Navier‐Stokes equations in boundary fitted co‐ordinates , 1992 .

[2]  Roger B. Sidje,et al.  Alternatives for parallel Krylov subspace basis computation , 1997, Numer. Linear Algebra Appl..

[3]  Erik Brakkee,et al.  A parallel domain decomposition algorithm for the incompressible Navier-Stokes equations , 1995, Simul. Pract. Theory.

[4]  Wei-Pai Tang,et al.  Generalized Schwarz Splittings , 1992, SIAM J. Sci. Comput..

[5]  Cornelis Vuik,et al.  Parallel implementation of a multiblock method with approximate subdomain solution , 1999 .

[6]  Olof B. Widlund,et al.  Domain Decomposition Algorithms with Small Overlap , 1992, SIAM J. Sci. Comput..

[7]  Joseph E. Pasciak,et al.  Analysis of non-overlapping domain decomposition algorithms with inexact solves , 1998, Math. Comput..

[8]  H. V. D. Vorst,et al.  Reducing the effect of global communication in GMRES( m ) and CG on parallel distributed memory computers , 1995 .

[9]  L. Reichel,et al.  A Newton basis GMRES implementation , 1994 .

[10]  H. Walker.  Implementation of the GMRES method using householder transformations , 1988 .

[11]  Pieter Wesseling,et al.  Domain decomposition for the incompressible Navier-Stokes equations: solving subdomain problems accu , 1995 .

[12]  Guangye Li,et al.  A Block Variant of the GMRES Method on Massively Parallel Processors , 1997, Parallel Comput..

[13]  P. C. Robinson,et al.  A numerical study of various algorithms related to the preconditioned conjugate gradient method , 1985 .

[14]  Per Weinerfelt,et al.  Coarse grid correction scheme for implicit multiblock Euler calculations , 1995 .

[15]  Cornelis Vuik,et al.  Numerical solution of the incompressible Navier-Stokes equations by Krylov subspace and multigrid methods , 1995, Adv. Comput. Math..

[16]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[17]  Wei Shyy,et al.  A pressure-based composite grid method for the Navier-Stokes equations , 1993 .

[18]  Pieter Wesseling,et al.  Computing Flows on General Three-Dimensional Nonsmooth Staggered Grids , 1999 .

[19]  Jinchao Xu,et al.  Some Nonoverlapping Domain Decomposition Methods , 1998, SIAM Rev..

[20]  J. Kan.  A second-order accurate pressure correction scheme for viscous incompressible flow , 1986 .

[21]  Cornelis Vuik,et al.  GMRESR: a family of nested GMRES methods , 1994, Numer. Linear Algebra Appl..

[22]  G. de Vahl Davis,et al.  Natural convection in a square cavity: A comparison exercise , 1983 .

[23]  O. Axelsson,et al.  On the rate of convergence of the preconditioned conjugate gradient method , 1986 .

[24]  L. Adleman,et al.  A Subexponential Algorithm for Discrete Logarithms over All Finite Fields , 1993, Annual International Cryptology Conference.

[25]  Cornelis Vuik,et al.  Parallelism in ILU-preconditioned GMRES , 1998, Parallel Comput..

[26]  Jocelyne Erhel,et al.  A parallel GMRES version for general sparse matrices. , 1995 .

[27]  Cornelis Vuik,et al.  FAST ITERATIVE SOLVERS FOR THE DISCRETIZED INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[28]  H. Bijl,et al.  Computing Flow on General Two-Dimensional Nonsmooth Staggered Grids , 1998 .

[29]  William Gropp,et al.  A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems , 1994, Numer. Linear Algebra Appl..

[30]  C. Vuik.  Solution of the discretized incompressible Navier‐Stokes equations with the GMRES method , 1993 .

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

[32]  Hester Bijl,et al.  A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates , 1998 .