Beyond pressureless gas dynamics: Quadrature-based velocity moment models

Following the seminal work of F. Bouchut on zero pressure gas dynamics, extensively used for gas particle-flows, the present contribution investigates quadrature-based velocity moments models for kinetic equations in the framework of the infinite Knudsen number limit, that is, for dilute clouds of small particles where the collision or coalescence probability asymptotically approaches zero. Such models define a hierarchy based on the number of moments and associated quadrature nodes, the first level of which leads to pressureless gas dynamics. We focus in particular on the four moment model where the flux closure is provided by a two-node quadrature in the velocity phase space and provides the right framework for studying both smooth and singular solutions. The link with both the kinetic underlying equation as well as with zero pressure gas dynamics, i.e. the dynamics at the frontier of the moment space of order four, is provided. We define the notion of measure solutions and characterize the mathematical structure of the resulting system of four PDEs. We exhibit a family of entropies and entropy fluxes and define the notion of entropic solution. We study the Riemann problem and provide entropic solutions in particular cases. This leads to a rigorous link with the possibility of the system of macroscopic PDEs to allow particle trajectory crossing (PTC) in the framework of smooth solutions. Generalized δ-shock solutions resulting from the Riemann problem are also investigated. Finally, using a kinetic scheme proposed in the literature in several areas, we validate such a numerical approach and propose a dedicated extension at the frontier of the moment space in the framework of both regular and singular solutions. This is a key issue for application fields where such an approach is extensively used.