Ordering Effects on Relaxation Methods Applied to the Discrete Convection-Diffusion Equation

  • Howard C. Elman
  • Michael P. Chernesky
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 60)


We present an analysis of relaxation methods for the discrete convection-diffusion equation based on norms of the iteration matrices. For one-dimensional problems, the results show how the performance of iterative solvers is affected by directions of flow associated with the underlying operator. In particular, for problems of size n, relaxation sweeps opposite the direction of flow incur a latency of approximately n steps in which convergence is slow, and red-black relaxation incurs a latency of approximately n/2 steps. There is no latency associated with relaxation that follows the flow. The one-dimensional analysis is also generalized to two-dimensional problems in the case where relaxation follows the flow.


Relaxation Method Iteration Matrix Centered Finite Difference Iteration Matrice Piecewise Linear Finite Element 
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  1. [1]
    J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  2. [2]
    M. Eiermann, Fields of Values and Iterative Methods, tech. report, IBM Scientific Center, Heidelberg, 1991.Google Scholar
  3. [3]
    H. C. Elman and M. P. Chernesky, Ordering effects on relaxation methods applied to the discrete convection-diffusion equation, Tech. Report UMIACS- TR-92-40, Institute for Advanced Computer Studies, University of Maryland, 1992.Google Scholar
  4. [4]
    H. C. Elman and G. H. Golub, Iterative methods for cyclically reduced non-self- adjoint linear systems, II, Math. Comp., 56 (1991), pp. 215–242.MathSciNetMATHGoogle Scholar
  5. [5]
    H. C. Elman and G. H. Golub, Line iterative methods for cyclically reduced convection-diffusion problems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 339–363.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    P. A. Farrell, Flow conforming iterative methods for convection dominated flows, in Numerical and Applied Mathematics, C. Brezinski, ed., J. C. Baltzer AG, Scientific Publishing Co., 1989, pp. 681–686.Google Scholar
  7. [7]
    P. M. Gresho and R. L. Lee, Don’t suppress the wiggles - they’re telling you something, Computers and Fluids, 9 (1981), pp. 223–253.MathSciNetCrossRefGoogle Scholar
  8. [8]
    C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, New York, 1987.MATHGoogle Scholar
  9. [9]
    L. N. Trefethen, Pseudospectra of Matrices, Tech. Report 91-10, Oxford University Computing Laboratory, 1991.Google Scholar
  10. [10]
    R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.Google Scholar
  11. [11]
    D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1970.Google Scholar

Copyright information

© Springer-Verlag New York, Inc 1994

Authors and Affiliations

  • Howard C. Elman
    • 1
    • 2
  • Michael P. Chernesky
    • 3
  1. 1.Department of Computer ScienceUniversity of MarylandCollege ParkUSA
  2. 2.Institute for Advanced Computer StudiesUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsUniversity of MarylandCollege ParkUSA

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