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{\large Chao Yang \\
{\bf Parallel Two-level Schwarz Methods for Fully Implicit Solution of Shallow Water Equations on the Cubed-sphere}}

Department of Computer Science \\ University of Colorado at Boulder \\ Boulder \\ CO 80309
\\
{\tt chao.yang@colorado.edu}\\
Xiao-Chuan Cai\\
{\em Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309}\end{center}

In this talk a parallel two-level Schwarz preconditioner is proposed as
part of a fully implicit solver for the shallow water equations
discretized with a finite volume scheme on the cubed-sphere. With such an
implicit time integration scheme the time step size is no longer
constrained by the CFL condition, which is usually required when using
explicit or semi-implicit methods. The price to pay is that a large
sparse nonlinear system of equations has to be solved at every time step.
When the one-level method is used, our numerical experiments show that
the number of nonlinear iterations per time step is almost independent of
the number of processors and the time step size, but the number of linear
iterations grows when we increase the number of processors. In this work
we introduce a coarse grid correction to the Schwarz preconditioner, and
we show numerically that, with the help of the second level, the number
of linear iterations is nearly independent of the number of processors
and is less sensitive to the implicit time step size. We present
numerical results obtained on machines with thousands of processors.


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