  Stochastic Galerkin formulations of the two-dimensional shallow water systems
parameterized with random variables may lose hyperbolicity, and hence change
the nature of the original model. In this work, we present a
hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting
the polynomial chaos approximations to the nonlinear terms of $(q^x)^2/h,
(q^y)^2/h$, and $(q^xq^y)/h$ in the shallow water equations. We derive a
sufficient condition to preserve the hyperbolicity of the stochastic Galerkin
system which requires only a finite collection of positivity conditions on the
stochastic water height at selected quadrature points in parameter space. Based
on our theoretical results for the stochastic Galerkin formulation, we develop
a corresponding well-balanced hyperbolicity-preserving central-upwind scheme.
We demonstrate the accuracy and the robustness of the new scheme on several
challenging numerical tests.
