Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations

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