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In this article we present an a posteriori error estimator for thespatial-stochastic error of a Galerkin-type discretisation of an initial valueproblem for a random hyperbolic conservation law. For the stochasticdiscretisation we use the Stochastic Galerkin method and for thespatial-temporal discretisation of the Stochastic Galerkin system a Runge-KuttaDiscontinuous Galerkin method. The estimator is obtained using smoothreconstructions of the discrete solution. Combined with the relative entropystability framework of Dafermos \cite{dafermos2005hyperbolic}, this leads tocomputable error bounds for the space-stochastic discretisation error.  Moreover, it turns out that the error estimator admits a splitting into onepart representing the spatial error, and a remaining term, which can beinterpreted as the stochastic error. This decomposition allows us to balancethe errors arising from spatial and stochastic discretisation. We conclude withsome numerical examples confirming the theoretical findings.

A posteriori error analysis for random scalar conservation laws using the Stochastic Galerkin method