  In this paper we extensively study the stochastic Galerkin scheme for
uncertain systems of conservation laws, which appears to produce oscillations
already for a simple example of the linear advection equation with Riemann
initial data. Therefore, we introduce a modified scheme that we call the
weighted essentially non-oscillatory (WENO) stochastic Galerkin scheme, which
is constructed to prevent the propagation of Gibbs phenomenon into the
stochastic domain by applying a slope limiter in the stochasticity. In order to
achieve a high order method, we use a spatial WENO reconstruction and also
compare the results to a scheme that uses WENO reconstruction in both the
physical and the stochastic domain. We evaluate these methods by presenting
various numerical test cases where we observe the reduction of the total
variation compared to classical stochastic Galerkin.
