The Effects of Dissipation and Coarse grid Resolution
for Multigrid in Flow Problems

                                Peter Eliasson
             FFA, Aeronautical Research Institute, Bromma, Sweden

                                     and

                                Bjorn Engquist
                                     KTH,
              Royal Institute of Technology, Stockholm, Sweden;
                            UCLA, California, USA

                                   Abstract

The objective of this paper is to investigate the convergence properties of
the Euler equation approximations.  The convergence is accomplished by
multi-stage explicit time-stepping to steady state combined with FAS multigrid
for different spatial discretizations.

The theoretical investigation is carried out for linear hyperbolic equations
in one and two dimensions.  Using a Fourier decomposition, special attention
is paid to the low and high frequencies.  The spectra reveals that for
stability and hence robustness of spatial discretizations with a small amount
of numerical dissipation the grid transfer operators have to be accurate
enough and the smoother of low temporal accuracy.

Numerical results give grid independent convergence in one dimension.  This is
proved in special cases.  For two-dimensional problems with a small amount of
numerical dissipation, however, only a few grid levels contribute to an
increased speed of convergence.  This is explained by the small numerical
dissipation leading to dispersion.  Increasing the mesh density and hence
making the problem over resolved increases the number of mesh levels
contributing to an increased speed of convergence.  If the steady state
equations are elliptic, all grid levels contribute to the convergence
regardless of the mesh density.  2D subsonic mixed hyperbolic/elliptic and
supersonic hyperbolic Euler computations confirm the results.

  References
[1] Eliasson, P. (1993) "Dissipation Mechanisms and Multigrid Solutions in a
    Multiblock Solver for Compressible Flow", Doctoral Thesis, TRITA-NA-R9314,
    ISSN-0348-2952.