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.

翻译：浅水流模型广泛用于描述河流、湖泊和沿海地区的水流运动。针对相应输运主导的非线性偏微分方程模型中不确定性的处理，在理论和数值计算层面均构成挑战，这也正是本文的核心创新点。从参数化/不确定性浅水方程的空间一维双曲保持、正性保持随机Galerkin格式出发，我们推导了该系统的熵-熵流对。利用这一熵-熵流对，为随机Galerkin浅水系统构建了二阶能量守恒保形格式，以及一阶和二阶能量稳定有限体积格式。通过多个数值实验验证了该方法的表现性能。