Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicity-preserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.
翻译:带有随机变量参数的二维浅水系统的Stochatic Galerkin 配方配方可能丧失超偏差,从而改变原始模型的性质。 在这项工作中,我们通过仔细选择浅水方程中非线性词的多偏差近似值,呈现出超偏差保存随机高温的Galerkin 配方。 我们获得足够的条件来保持高温浅水系统的双偏差性,它只需要在参数空间中选定等离子点对热水高度的假设条件进行有限的收集。 根据我们关于高压加勒金配方的理论结果,我们开发了相应的平衡的超偏差保全中上风方案。我们在若干具有挑战性的数字测试中展示了新方案的准确性和稳健性。