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{\large Chao Yang \\
{\bf Multilevel Schwarz Preconditioner for a Fully Implicit Global Shallow Water Solver}}

Department of Computer Science \\ University of Colorado at Boulder \\ Boulder \\ CO 80309-0430
\\
{\tt yangchao.china@gmail.com}\\
Xiao-Chuan Cai\end{center}

A multilevel Schwarz preconditioner is proposed as part of a fully
implicit solver for the shallow water equations discretized with a
second-order well-balanced finite volume scheme on the cubed-sphere.
Fully implicit methods have the advantage that the time step size is no
longer constrained by the CFL condition. But a large sparse nonlinear
system of equations has to be solved at every time step. To solve these
nonlinear systems, we use a Newton-Krylov-Schwarz method, in which the
multilevel Schwarz preconditioner plays a key role for the effectiveness
and efficiency. We show by numerical experiments that with the multilevel
method, the numbers of nonlinear and linear iterations per time step are
almost independent on the number of processors and much less sensitive to
the implicit time step size compared to the one-level method. Numerical
results obtained on machines with thousands of processors are provided to
show the parallel scalability of the solver.


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