Abstract
Statistical solutions have recently been introduced as 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 obtained from the Runge-Kutta Discontinuous Galerkin method in one spatial dimension, 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.
Highlights
Analysis and numerics of hyperbolic conservation laws have seen a significant shift of paradigms in the last decade
The work at hand tries to complement the a priori analysis from [17] with a reliable a posteriori error estimator, i.e., we propose a computable upper bound for the numerical approximation error of statistical solutions
Our numerical approximations rely on so-called regularized empirical measures, which enable us to use the relative entropy method of Dafermos and DiPerna [8] within the framework of dissipative statistical solutions introduced by the authors of [17]
Summary
Analysis and numerics of hyperbolic conservation laws have seen a significant shift of paradigms in the last decade. For systems in multiple space dimensions the non-uniqueness of entropy weak solutions immediately implies non-uniqueness of dissipative measure valued solutions and statistical solutions. Still, all these concepts satisfy weakstrong uniqueness principles, i.e., as long as a Lipschitz continuous solution exists in any of these classes it is the unique solution in any of these classes. The work at hand tries to complement the a priori analysis from [17] with a reliable a posteriori error estimator, i.e., we propose a computable upper bound for the numerical approximation error of statistical solutions This extends results for entropy weak solutions of deterministic and random systems of hyperbolic conservation laws [10, 20, 21, 30] towards statistical solutions.
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