Honoring Ted O’Brien: High order methods for filtered and probability density function models

The systems of partial differential equations that govern probability and filtered density function models can be solved directly using numerical methods. Oftentimes, this type of system is also solved using a combination of Monte-Carlo and stochastic differential equations. If the density function model is coupled with another model that has feedback, as can occur in multi-physics or multi-phase environments, then the numerical coupling must be consistent for both approaches to obtain an accurate numerical solution. In this talk, I will discuss recent progress in the development of high order accuracy methods for models governing, chemically reaction and/or particle-laden, high-speed flows with shocks. High order distribution functions and weighted interpolations combined with spectral elements methods are presented and are shown to give accurate results for time-dependent problems that require long time integration.