Error control for statistical solutions

TL;DR

This paper introduces a new a posteriori error estimate for statistical solutions of hyperbolic conservation laws, combining deterministic and stochastic error components, validated through numerical experiments.

Contribution

It develops a novel Wasserstein distance-based error estimator for dissipative statistical solutions and their numerical approximations, including a split into deterministic and stochastic errors.

Findings

Error estimator effectively captures approximation errors.

Numerical experiments confirm the splitting of residuals.

Scaling properties of residuals are verified.

Abstract

Statistical solutions have recently been introduced as a an alternative solution framework for hyperbolic systems of conservation laws. In this work we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their splitting.