In this article we consider one-dimensional random systems of
hyperbolic conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws which
involve random initial data and random flux functions.
Based on these results we present an a posteriori error analysis for a numerical approximation
of the random entropy solution.
For the stochastic discretization, we consider a non-intrusive approach, the
Stochastic Collocation method. The spatial-temporal discretization
relies on the Runge--Kutta Discontinuous Galerkin method.
We derive the a posteriori estimator
using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework
this yields computable error bounds for the entire space-stochastic discretization error.
The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing
for a novel residual-based space-stochastic adaptive mesh refinement algorithm.
We conclude with various numerical examples investigating the scaling properties of
the residuals and
illustrating the efficiency of the proposed adaptive algorithms.