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

TL;DR

This paper introduces a novel hyperbolicity-preserving stochastic Galerkin method for 1D shallow water equations, ensuring stability and accuracy in uncertain environments through polynomial chaos expansion and a well-balanced scheme.

Contribution

It develops a new stochastic Galerkin formulation that maintains hyperbolicity and introduces a well-balanced scheme for the stochastic shallow water equations, with conditions guaranteeing hyperbolicity.

Findings

Method successfully preserves hyperbolicity in numerical tests

Achieves stable and accurate solutions for stochastic shallow water problems

Provides conditions to ensure hyperbolicity at stochastic quadrature points

Abstract

A stochastic Galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve hyperbolicity of the original system. In this work, we develop 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 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 hyperbolicty-preserving CFL-type condition. The performance of the developed method is illustrated on a number of challenging numerical tests.