An a posteriori error analysis based on non-intrusive spectral projections for systems of random conservation laws

TL;DR

This paper develops an a posteriori error estimator for one-dimensional random hyperbolic systems of conservation laws, combining spectral projection in the random space with Runge-Kutta Discontinuous Galerkin methods for space-time discretization.

Contribution

It introduces a novel a posteriori error analysis framework that provides computable bounds and separates stochastic and deterministic error components.

Findings

Derives an error estimator using smooth reconstructions and relative entropy stability.

Provides a splitting of the error estimator into stochastic and deterministic parts.

Establishes bounds for space-stochastic discretization error.

Abstract

We present an a posteriori error analysis for one-dimensional random hyperbolic systems of conservation laws. For the discretization of the random space we consider the Non-Intrusive Spectral Projection method, the spatio-temporal discretization uses the Runge--Kutta Discontinuous Galerkin method. We derive an a posteriori error estimator using smooth reconstructions of the numerical solution, which combined with the relative entropy stability framework yields computable error bounds for the space-stochastic discretization error. Moreover, we show that the estimator admits a splitting into a stochastic and deterministic part.