Determination of the probability of randomly forced point-particle tracers

A method is proposed that determines the probability of the location and velocity of a randomly forced, Lagrangian point-particle trace. Randomness naturally describes deviations from the exact Stokes drag law of a spherical shaped point-particle for different shapes and physical conditions. A hyperbolic PDE model is derived that propagates this randomness into its joint PDF solution that governs the particle phase. A Method of Characteristics (MoC) solves the PDE locally and thus determines the probability of a single trace. The traces seeded at a tensorial grid of Jacobi quadrature points at initial time and computed over a finite time yield a high-order flow map. This flow map is used to determine marginals and moments of the PDF. The MoC approach is not subject to grid based numerical inaccuracies and instabilities in the solution of Eulerian form such as spatial approximation errors and Gibbs oscillations. We validate this novel framework with several tests and show that the MoC is accurate and computationally more efficient than the Eulerian method and Monte Carlo (MC) based methods.