Error control for statistical solutions of hyperbolic systems of conservation laws
The paper deals with hyperbolic systems of conservation laws and provides a reliable a posteriori error estimator, i.e., a computable upper bound for the numerical approximation error of dissipative statistical solutions is proposed in one spatial dimension. To this end, regularized empirical measures are used and the so-called relative entropy method by \textit{C. M. Dafermos} and \textit{R. J. DiPerna} [J. Differ. Equations 20, 90--114 (1976; Zbl 0323.35050)] can be applied. The error estimator is split into a stochastic and a spatiotemporal part. The Wasserstein distance between dissipative statistical solutions and the numerical approximation computed with a Runge-Kutta DG method in one spatial dimension is determined. Different numerical experiments illustrate the theoretical results, including a numerical approximation of the Wasserstein distance and an application to smooth and non-smooth solutions.
scientific article; zbMATH DE number 7378109
scientific article