What are we computing with numerical methods for hyperbolic systems of conservation laws?

Efficient numerical methods for approximating hyperbolic systems of conservation laws have been in existence for the past three decades. However, rigorous convergence results to entropy solutions are only available in the case of scalar conservation laws. We present numerical evidence that demonstrates the lack of convergence of state of art numerical methods to entropy solutions of multi-dimensional systems. On the other hand, an ensemble averaged version of these numerical methods is shown to converge to entropy measure-valued solutions. However, these solutions are not unique. We impose additional admissibility criteria by requiring propagation of information on all multi-point correlations. This results in the concept of statistical solutions or time-parametrized probability measures on integrable functions, as a solution framework. We derive sufficient conditions for convergence of ensemble-averaged numerical methods to statistical solutions and present numerical experiments illustrating these solutions.