Parallelism in GMRES applied to the computation of incompressible flows

A parallel implementation of the preconditioned GMRES method is described. The method is applied to solve the discretized incompressible Navier-Stokes equations. A parallel implementation of the inner product is given, which appears to be scalable on a massively parallel computer. The most diicult part to parallelize is the ILU-preconditioner. We parallelize the preconditioner using ideas proposed by Bastian and Horton 2]. Their ideas are little used to parallelize preconditioned Krylov subspace methods. Contrary to other parallel methods the required number of iterations is independent of the number of processors used. A model is presented to predict the eeciency of the method. Experiments are done on the Cray T3D computing the solution of a two-dimensional ow. Predictions of computing time show good correspondence with measurements. It appears that the proposed method is scalable.

[1]  Marcel Zijlema,et al.  ON THE CONSTRUCTION OF A THIRD‐ORDER ACCURATE MONOTONE CONVECTION SCHEME WITH APPLICATION TO TURBULENT FLOWS IN GENERAL DOMAINS , 1996 .

[2]  P. Wesseling,et al.  A numerical method for the computation of compressible flows with low Mach number regions , 1996 .

[3]  P. H. Michielse.  High Performance Computing: Trends and Expectations , 1996 .

[4]  Cornelis Vuik,et al.  Some parallelizable preconditioners for GMRES , 1996 .

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

[6]  Marcel Zijlema,et al.  Finite volume computation of incompressible turbulent flows in general co‐ordinates on staggered grids , 1995 .

[7]  P. Wesseling,et al.  Multigrid solution of the incompressible Navier-Stokes equations in general coordinates , 1994 .

[8]  Guus Segal,et al.  On the implementation of symmetric and antisymmetric periodic boundary conditions for incompressible flow , 1994 .

[9]  M. J. Quinn,et al.  Parallel Computing: Theory and Practice , 1994 .

[10]  P. Wesseling,et al.  Benchmark solutions for the incompressible Navier–Stokes equations in general co‐ordinates on staggered grids , 1993 .

[11]  A. L. Couch,et al.  Locating performance problems in massively parallel executions , 1993, Proc. IEEE.

[12]  Cornelis W. Oosterlee,et al.  Multigrid Schemes for Time-Dependent Incompressible Navier-Stokes Equations , 1993, IMPACT Comput. Sci. Eng..

[13]  Cornelis W. Oosterlee,et al.  Steady incompressible flow around objects in general coordinates with a multigrid solution method , 1994 .

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

[15]  Graham Horton,et al.  Parallelization of Robust Multigrid Methods: ILU Factorization and Frequency Decomposition Method , 1991, SIAM J. Sci. Comput..

[16]  P. Wesseling,et al.  Finite volume discretization of the incompressible Navier-Stokes equations in general coordinates on staggered grids , 1991 .

[17]  Jack J. Dongarra,et al.  Solving linear systems on vector and shared memory computers , 1990 .

[18]  Robert E. Benner,et al.  Development of Parallel Methods for a $1024$-Processor Hypercube , 1988 .

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

[20]  R. Grimes,et al.  On vectorizing incomplete factorization and SSOR preconditioners , 1988 .

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

[22]  Å. Björck.  Solving linear least squares problems by Gram-Schmidt orthogonalization , 1967 .