  A stochastic Galerkin formulation for a stochastic system of balanced or
conservation laws may fail to preserve hyperbolicity of the original system. In
this work, we develop hyperbolicity-preserving stochastic Galerkin formulation
for the one-dimensional shallow water equations by carefully selecting the
polynomial chaos expansion of the nonlinear $q^2/h$ term in terms of the
polynomial chaos expansions of the conserved variables. In addition, in an
arbitrary finite stochastic dimension, we establish a sufficient condition to
guarantee hyperbolicity of the stochastic Galerkin system through a finite
number of conditions at stochastic quadrature points. Further, we develop a
well-balanced central-upwind scheme for the stochastic shallow water model and
derive the associated hyperbolicty-preserving CFL-type condition. The
performance of the developed method is illustrated on a number of challenging
numerical tests.
