Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Abstract

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 of (q x ) 2 /h, (q y ) 2 /h, and (q x q y )/h 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 hyperbolicitypreserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

Description of random level sets by polynomial chaos expansions

“…In particular the capacity form is proven strongly hyperbolic. The proof is inspired by the approach in [7,8], where the Jacobian is shown to be similar to a symmetric matrix.…”

“…The drawback of introducing auxiliary variables is an additional computational overhead that arises from an optimization problem, which is required to calculate the auxiliary variables. Recently, a hyperbolic stochastic Galerkin for the shallow water equations has been presented that neither requires auxiliary variables nor any transform, since the Jacobian is shown to be similar to a symmetric matrix [7,8]. Furthermore, Hamilton-Jacobi equations have been solved by using the Jin-Xin relaxation [19].…”

“…To establish hyperbolicity for the level-set equations in multiple dimensions with uncertainties in velocities, we follow a strategy that is based both on the introduction of auxiliary variables and on the symmetrization of the Jacobian [7,8]. First, an expression for the Euclidean norm is presented.…”

A realizable filtered intrusive polynomial moment method

“…Compared to other intrusive methods such as the intrusive polynomial moment (IPM) method [6] or the Roe transformation method [7], stochastic-Galerkin is computationally inexpensive, however suffers from a possible loss of hyperbolicity [6]. This necessitates manipulating the solution to be able to run the method in certain settings [8] or imposing a modification of the flux discretization [9]. Moreover, SG suffers from oscillations when the solution is not sufficiently smooth [10], which is a common issue in hyperbolic conservation laws where the solution tends to form discontinuous shocks [6].…”

Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function

“…Recently, a hyperbolic stochastic Galerkin formulation for the shallow water equations has been presented that neither requires auxiliary variables nor any transform, since the Jacobian is shown to be similar to a symmetric matrix [12,13]. Then, positivity of the water height at a finite number of stochastic quadrature points is sufficient to preserve hyperbolicity of the stochastic Galerkin formulation.…”