From Phase Velocity Analysis to TVD Multigrid for Convection
       Dominated Problems
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Abstract:
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Efficient solution of convection dominated problems by multigrid methods
is challenging since algorithms and techniques tend to be very different
in the two limiting cases of elliptic and hyperbolic equations.
Different approaches and techniques have been proposed and developed to
cope with the difficulties, for instance, modified Runge-Kutta and
Gauss-Seidel smoothings, operator-dependent interpolations, higher order
coarse grid corrections. In this talk, we study the efficiency of some
of these approaches based on the framework of phase velocity analysis
which provides more insight into the wave propagation in the cross- and
characteristic directions, especially between coarse and fine grids,
in supplement to the classical multigrid analysis. In particular, we
consider three commonly used coarse grid correction methods which show
similar elliptic multigrid convergence but quite different convergence
for hyperbolic equations due to their distinct phase velocity errors.
Finally, we demonstrate the importance of the phase velocity
(or dispersion) concept in the new design of robust algorithms by an
efficient total variation diminishing multigrid method.