Beyond pressureless gas dynamics: Quadrature-based velocity moment\n models

Abstract

Following the seminal work of F. Bouchut on zero pressure gas dynamics which\nhas been extensively used for gas particle-flows, the present contribution\ninvestigates quadrature-based velocity moments models for kinetic equations in\nthe framework of the infinite Knudsen number limit, that is, for dilute clouds\nof small particles where the collision or coalescence probability\nasymptotically approaches zero. Such models define a hierarchy based on the\nnumber of moments and associated quadrature nodes, the first level of which\nleads to pressureless gas dynamics. We focus in particular on the four moment\nmodel where the flux closure is provided by a two-node quadrature in the\nvelocity phase space and provide the right framework for studying both smooth\nand singular solutions. The link with both the kinetic underlying equation as\nwell as with zero pressure gas dynamics is provided and we define the notion of\nmeasure solutions as well as the mathematical structure of the resulting system\nof four PDEs. We exhibit a family of entropies and entropy fluxes and define\nthe notion of entropic solution. We study the Riemann problem and provide a\nseries of entropic solutions in particular cases. This leads to a rigorous link\nwith the possibility of the system of macroscopic PDEs to allow particle\ntrajectory crossing (PTC) in the framework of smooth solutions. Generalized\n$\\delta$-choc solutions resulting from Riemann problem are also investigated.\nFinally, using a kinetic scheme proposed in the literature without mathematical\nbackground in several areas, we validate such a numerical approach in the\nframework of both smooth and singular solutions.\n