Finite volume methods for the computation of statistical solutions of the incompressible Euler equations

Abstract: We present an efficient numerical scheme based on Monte Carlo integration to approximate statistical solutions of the incompressible Euler equations. The scheme is based on finite volume methods, which provide a more flexible framework than previously existing spectral methods for the computation of statistical solutions for incompressible flows. This finite volume scheme is rigorously proven to, under experimentally verifiable assumptions, converge in an appropriate topology and with increasing resolution to a statistical solution. The convergence obtained is stronger than that of measure-valued solutions, as it implies convergence of multi-point correlation marginals. We present results of numerical experiments which support the claim that the aforementioned assumptions are very natural, and appear to hold in practice.