  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.
