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

In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial-temporal discretization of the stochastic Galerkin system a Runge-Kutta discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016, Hyperbolic Conservation Laws in Continuum Physics, 4th edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space-stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.