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

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.