Statistical solutions of hyperbolic systems of conservation laws: numerical approximation

Error control for statistical solutions

“…This has led to several weaker (more statistics inspired) notions of solutions being proposed. We would like to mention dissipative measure valued solutions [12] and statistical solutions [16]. Considering measure valued solutions has a long history in hyperbolic conservation laws and can be traced back to the works of DiPerna [11] considering Young measures.…”

“…Considering measure valued solutions has a long history in hyperbolic conservation laws and can be traced back to the works of DiPerna [11] considering Young measures. Statistical solutions are time-parametrized probability measures on spaces of integrable functions and have been introduced recently for scalar problems in [14] and for systems in [16]. We would like to mention that in the context of the incompressible Navier-Stokes equations and turbulence, statistical solutions have a long history going back to the seminal work of Foias et al, see [18] and references therein.…”

“…The precise definition of statistical solutions is based on an equivalence theorem [16,Theorem 2.8] that relates probability measures on spaces of integrable functions and correlation measures on the state space. Correlation measures are measures that determine joint probability distributions of some unknown quantity at any finite collection of spatial points.…”

“…Still, all these concepts satisfy weak-strong 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 (technical) basis for obtaining weak-strong uniqueness results is based on the relative entropy framework of Dafermos and DiPerna [8], which can be extended to dissipative statistical solutions as in [16].…”

“…The work at hand tries to complement the a priori analysis from [16] 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] towards statistical solutions.…”