Conditional Quadrature Method of Moments for Kinetic Equations

Kinetic equations arise in a wide variety of physical systems and efficient numerical methods are needed for their solution. Moment methods are an important class of approximate models derived from kinetic equations, but require closure to truncate the moment set. In quadrature-based moment methods, closure is achieved by inverting a finite set of moments to reconstruct a point distribution from which all unclosed moments (e.g. spatial fluxes) can be related to the finite moment set. In this work, a novel moment-inversion algorithm, based on 1-D quadrature of conditional velocity moments, is introduced and shown to always yield realizable distribution functions (i.e. non-negative quadrature weights). Then Realizable High-Order Finite-Volume Schemes (Vikas, et al., 2010) are applied to this conditional quadrature method of moments (CQMOM) to increase accuracy. The CQMOM can be used to compute exact N-point quadratures for multi-valued solutions (also known as the multi-variate truncated moment problem), and provides optimal approximations of continuous distributions. In order to control numerical errors arising in spatial transport, an adaptive 1-D quadrature algorithm is formulated for use with CQMOM. The use of adaptive CQMOM for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing (i.e. collision-less systems), elastic and inelastic particle-particle collisions, and external forces (i.e. fluid drag).