SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A929-A952, January 2021.
A stochastic Galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve the hyperbolicity of the original system. In this work, we develop a 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 the 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 hyperbolicity-preserving CFL-type condition. The performance of the developed method is illustrated on a number of challenging numerical tests.

中文翻译：

---

浅水方程组的双曲保性和均衡的随机Galerkin方法

SIAM科学计算杂志，第43卷，第2期，第A929-A952页，2021年1月。
平衡或守恒定律的随机系统的随机Galerkin公式可能无法保留原始系统的双曲性。在这项工作中，我们通过根据守恒变量的多项式混沌展开式仔细选择非线性$ q ^ 2 / h $项的多项式混沌展开式，为一维浅水方程式开发了一个保持双曲率的随机Galerkin公式。 。另外，在任意有限的随机维度上，我们建立了一个充分的条件，以通过在随机正交点上的有限数量的条件来保证随机Galerkin系统的双曲性。此外，我们为随机浅水模型开发了一个平衡良好的中心迎风方案，并推导了相关的保双曲CFL类型条件。