Speaker
Description
The statistical behaviors of hyperbolic conversation laws with random initial data and/or stochastic forcing terms are crucial for uncertainty quantification and deeper insights into turbulent flows. We start from derivations of known hierarchies governing the evolution of probability distribution functions f, and propose a new set of hierarchies relating locally higher-order spatial derivatives. For such kinetic-like equations, a viscosity-induced unclosed term remains, which is demonstrated to play a central role in preserving the positivity of f in the scalar case. Illustrating examples will be discussed to shed light on the properties of the unclosed term. For general cases, closure through the piecewise solution of the underlying conservation laws is desirable. To mediate the computational difficulties raised from the viscosity term with second-order spatial derivatives, we propose a novel first-order relaxation system to approximate the incompressible Navier-Stokes equations, and prove the convergence of the solutions to those of the limit system when two relaxation parameters both approaches zero. The new system is promising for our multiscale numerical pproach for the statistical behaviors, and this is our ongoing work under the project SPP 24-10.
