  The shallow water flow model is widely used to describe water flows in
rivers, lakes, and coastal areas. Accounting for uncertainty in the
corresponding transport-dominated nonlinear PDE models presents theoretical and
numerical challenges that motivate the central advances of this paper. Starting
with a spatially one-dimensional hyperbolicity-preserving,
positivity-preserving stochastic Galerkin formulation of the
parametric/uncertain shallow water equations, we derive an entropy-entropy flux
pair for the system. We exploit this entropy-entropy flux pair to construct
structure-preserving second-order energy conservative, and first- and
second-order energy stable finite volume schemes for the stochastic Galerkin
shallow water system. The performance of the methods is illustrated on several
numerical experiments.
