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

TL;DR

This paper develops an a posteriori error estimator for the combined spatial and stochastic discretisation errors in solving random hyperbolic conservation laws using the Stochastic Galerkin method and Runge-Kutta Discontinuous Galerkin schemes.

Contribution

It introduces a new error estimator that separates spatial and stochastic errors, enabling better error control in stochastic hyperbolic PDEs.

Findings

The error estimator provides computable bounds for discretisation errors.

The estimator effectively decomposes total error into spatial and stochastic components.

Numerical examples confirm the theoretical error bounds and decomposition.

Abstract

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.