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Efficient matrix decomposition for implicit algorithms

  • Robert W. MacCormack
Algorithms Navier-Stokes
Part of the Lecture Notes in Physics book series (LNP, volume 490)

Abstract

Implicit procedures for solving the equations of computational fluid dynamics have been developed and used widely during the past quarter century. They all attempt to solve a matrix equation formed by partial linearization and discretization of the governing fluid flow equations. The practical ones in use today for multidimensional problems first approximate the original matrix equation with a "pre-conditioned" matrix equation that can be solved efficiently. Two such procedures are SIP, or strongly Implicit Procedure for L-U decomposition, originally presented by Stone for solving the heat equation in two and three dimensions, and AF, or spacial Approximate Factorization, used centrally in the Beam-Warming and the Brily-McDonald methods. They each have distinct advantages and disadvantages. A small modification to either, to be presented herein, can produce a new factorization procedure containing the advantages of each and the disadvantages of neither.

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References

  1. 1.
    Stone,H.L., "Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations," SIAM J. Numer. Anal., Vol.5, No. 3, 1968Google Scholar
  2. 2.
    Hirsch,C., "Numerical Computation of Internal and External Flows," John Wiley, Vol. I, 1988, and Vol. II, 1990.Google Scholar
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    Chakravarthy,S.R., "Relaxation Methods for Upwind Unfactored Upwind Schemes," AIAA Paper No. 84-0165, 1984.Google Scholar
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    Hakkinen,R.J., Greber,I., Trilling,L., and Abarbanel,S.S., "The Interaction of an Oblique Shock Wave with a Laminar Boundary Layer," Memo 2-18-59W, 1959, NASA.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Robert W. MacCormack
    • 1
  1. 1.Department of Aeronautics and AstronauticsStanfordUSA

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