Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number

A fourth-order compact finite difference scheme is employed with the multigrid algorithm to obtain highly accurate numerical solution of the convection-diffusion equation with very high Reynolds number and variable coefficients. The multigrid solution process is accelerated by a minimal residual smoothing (MRS) technique. Numerical experiments are employed to show that the proposed multigrid solver is stable and yields accurate solution for high Reynolds number problems. We also show that the MRS acceleration procedure is efficient and the acceleration cost is negligible. © 1997 John Wiley & Sons, Inc.

[1]  Homer F. Walker,et al.  Residual smoothing and peak/plateau behavior in Krylov subspace methods , 1995 .

[2]  A. Reusken,et al.  Multigrid with Matrix-dependent Transfer Operators for Convection-diffusion Problems , 1994 .

[3]  P. Hemker,et al.  Mixed defect correction iteration for the accurate solution of the convection diffusion equation , 1982 .

[4]  Achi Brandt,et al.  On Recombining Iterants in Multigrid Algorithms and Problems with Small Islands , 1995, SIAM J. Sci. Comput..

[5]  G. Golub,et al.  ITERATIVE METHODS FOR CYCLICALLY REDUCED NON-SELF-ADJOINT LINEAR SYSTEMS , 1990 .

[6]  Homer F. Walker,et al.  Residual Smoothing Techniques for Iterative Methods , 1994, SIAM J. Sci. Comput..

[7]  I. Yavneh,et al.  On Multigrid Solution of High-Reynolds Incompressible Entering Flows* , 1992 .

[8]  Jun Zhang,et al.  Residual scaling techniques in multigrid, I: Equivalence proof , 1997 .

[9]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[10]  Murli M. Gupta,et al.  A single cell high order scheme for the convection‐diffusion equation with variable coefficients , 1984 .

[11]  S. Schaffer,et al.  Higher order multigrid methods , 1984 .

[12]  Achi Brandt,et al.  Inadequacy of first-order upwind difference schemes for some recirculating flows , 1991 .

[13]  N. G. Wright,et al.  An efficient multigrid approach to solving highly recirculating flows , 1995 .

[14]  G. Carey,et al.  High‐order compact scheme for the steady stream‐function vorticity equations , 1995 .

[15]  P. M. de Zeeuw,et al.  The Convergence Rate of Multi-Level Algorithms Applied to the Convection-Diffusion Equation , 1985 .

[16]  Murli M. Gupta.  A fourth-order poisson solver , 1984 .

[17]  A. Brandt.  Guide to multigrid development , 1982 .

[18]  Arnold Reusken,et al.  Fourier analysis of a robust multigrid method for convection-diffusion equations , 1995 .

[19]  P. M. De Zeeuw,et al.  Matrix-dependent prolongations and restrictions in a blackbox multigrid solver , 1990 .

[20]  S. Dennis,et al.  Compact h4 finite-difference approximations to operators of Navier-Stokes type , 1989 .

[21]  M. M. Gupta,et al.  Comparison of 2nd and 4th Order Discretizations for Multigrid Poisson Solvers , 1995 .

[22]  Long Chen.  INTRODUCTION TO MULTIGRID METHODS , 2005 .

[23]  Murli M. Gupta.  High accuracy solutions of incompressible Navier-Stokes equations , 1991 .

[24]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[25]  Irfan Altas,et al.  A high accuracy defect-correction multigrid method for the steady incompressible Navier-Stokes equations , 1994 .

[26]  Alan E. Berger,et al.  Generalized OCI schemes for boundary layer problems , 1980 .

[27]  Achi Brandt,et al.  Accelerated Multigrid Convergence and High-Reynolds Recirculating Flows , 1993, SIAM J. Sci. Comput..

[28]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[29]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[30]  Willi Schönauer,et al.  Scientific computing on vector computers , 1987, Special topics in supercomputing.

[31]  Wolfgang Hackbusch,et al.  On multi-grid iterations with defect correction , 1982 .

[32]  Rüdiger Weiss,et al.  Convergence behavior of generalized conjugate gradient methods , 1990 .

[33]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[34]  Jun Zhang,et al.  Minimal residual smoothing in multi-level iterative method , 1997 .

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

[36]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[37]  S. Schaffer,et al.  Higher Order Multi-Grid Methods* , 1984 .

[38]  W. Auzinger.  Defect corrections for multigrid solutions of the Dirichlet problem in general domains , 1987 .

[39]  B. Fornberg,et al.  A compact fourth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations , 1995 .