A hyperbolicity-preserving discontinuous stochastic Galerkin scheme for uncertain hyperbolic systems of equations

TL;DR

This paper introduces a modified stochastic Galerkin scheme with a slope limiter to preserve hyperbolicity in uncertain hyperbolic systems, demonstrating high accuracy and robustness especially with discontinuities, through analysis and numerical tests.

Contribution

It presents a hyperbolicity-preserving stochastic Galerkin method using a slope limiter and discontinuous Galerkin scheme, improving stability and accuracy for uncertain hyperbolic systems.

Findings

The scheme maintains hyperbolicity with high-order accuracy.

It effectively handles discontinuities in uncertain initial conditions.

Performance compares favorably with stochastic collocation in multiple stochastic dimensions.

Abstract

Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a modification of this method that uses a slope limiter to retain admissible solutions of the system, while providing high-order approximations in the physical and stochastic space. This is done using a spatial discontinuous Galerkin scheme and a Multi-Element stochastic Galerkin ansatz in the random space. We analyze the convergence of the resulting scheme and apply it to the compressible Euler equations with various uncertain initial states in one and two spatial domains with up to three uncertainties. The performance in multiple stochastic dimensions is compared to the non-intrusive Stochastic Collocation method. The numerical results underline the strength of our method, especially if discontinuities are present in the uncertainty of the solution.