A posteriori error analysis for random scalar conservation laws using the Stochastic Galerkin method
For researchers in numerical analysis and uncertainty quantification, this provides a computable error bound and error splitting for a specific class of problems, but the approach is incremental as it builds on existing frameworks.
The paper presents an a posteriori error estimator for the spatial-stochastic error in Galerkin-type discretizations of random hyperbolic conservation laws, enabling error decomposition and balancing between spatial and stochastic discretizations. Numerical examples confirm the theoretical results.
In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the Stochastic Galerkin method and for the spatial-temporal discretisation 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 \cite{dafermos2005hyperbolic}, this leads to computable error bounds for the space-stochastic discretisation 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 discretisation. We conclude with some numerical examples confirming the theoretical findings.