Weighted essentially non-oscillatory stochastic Galerkin approximation for hyperbolic conservation laws

•Study of Gibbs oscillations within the stochastic Galerkin scheme for uncertain conservation laws.•Discontinuities in the physical space propagate into the solution manifold.•Introduction of the weighted essentially non-oscillatory (WENO) stochastic Galerkin.•The scheme includes a WENO reconstruction and a stochastic slope limiter.•A comparison to classical stochastic Galerkin demonstrates the reduction of the total variation, i.e. the oscillations.