  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.
