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

TL;DR

This paper introduces a modified stochastic Galerkin method that preserves hyperbolicity in uncertain hyperbolic PDE systems, ensuring stability and accuracy in uncertainty quantification for models like Euler equations and radiative transfer.

Contribution

It develops a hyperbolicity-preserving modification of the stochastic Galerkin method using a slope limiter, enhancing stability and applicability in uncertain hyperbolic systems.

Findings

The method maintains hyperbolicity in uncertain hyperbolic PDEs.

Numerical results show competitive accuracy with existing UQ methods.

The approach is computationally inexpensive and easy to implement.

Abstract

Uncertainty Quantification through stochastic spectral methods is rising in popularity. We derive a modification of the classical stochastic Galerkin method, that ensures the hyperbolicity of the underlying hyperbolic system of partial differential equations. The modification is done using a suitable "slope" limiter, based on similar ideas in the context of kinetic moment models. We apply the resulting modified stochastic Galerkin method to the compressible Euler equations and the $M_1$ model of radiative transfer. Our numerical results show that it can compete with other UQ methods like the intrusive polynomial moment method while being computationally inexpensive and easy to implement.