A posteriori error control for statistical solutions of barotropic Navier-Stokes equations

This project addresses the numerical approximation of statistical solutions of the barotropic Navier-Stokes equations, one of the fundamental equations in fluid mechanics. Statistical solutions are a novel solution concept for compressible Navier-Stokes equations that is motivated by turbulence modeling and is thought to address issues with well-posedness that persist for deterministic solution concepts. Statistical solutions can be understood as time-parametrized  probability measures on function spaces induced by a random initial datum. Hence, a statistical solution can be approximated by an empirical measure obtained from samples from the initial distribution that are evolved with a numerical scheme for the deterministic, barotropic Navier-Stokes equations. In the convection-dominated case that we focus on, a typical numerical scheme would be of Runge-Kutta discontinuous Galerkin type. We aim to provide reliable, efficient, and robust a posteriori error estimators for these schemes, i.e., upper error bounds for errors caused by discretization in space-time and stochastic space that are computable from the numerical solution. We will combine the relative entropy stability framework with suitable reconstructions of the numerical solution to establish these error bounds. Furthermore, we plan to employ our a posteriori error estimator to construct adaptive, highly efficient multi-level Monte Carlo schemes for approximating quantities of interest pertinent to statistical solutions.