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hal-00535782, version 2

## Beyond pressureless gas dynamics : Quadrature-based velocity moment models

Christophe Chalons () 12, Damien Kah () 23, Marc Massot () 2

Communications in Mathematical Sciences 10, 4 (2012) 1241-1272

Abstract: Following the seminal work of F. Bouchut on zero pressure gas dynamics which has been 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 provide 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 is provided and we define the notion of measure solutions as well as 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 a series of 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 $\delta$-choc solutions resulting from Riemann problem are also investigated. Finally, using a kinetic scheme proposed in the literature without mathematical background in several areas, we validate such a numerical approach in the framework of both smooth and singular solutions.

• 1:  Laboratoire Jacques-Louis Lions (LJLL)
• CNRS : UMR7598 – Université Pierre et Marie Curie [UPMC] - Paris VI
• 2:  Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion (EM2C)
• CNRS : UPR288 – Ecole Centrale Paris
• 3:  IFP Energies Nouvelles (IFPEN)
• IFP Energies Nouvelles
• Domain : Mathematics/Numerical Analysis
Mathematics/Analysis of PDEs
Engineering Sciences/Reactive fluid environment
• Keywords : Quadrature-based moment methods – gas-particle flows – kinetic theory – Particle trajectory crossing – entropic measure solution
• Available versions :  v1 (2010-11-12) v2 (2012-03-20)

• hal-00535782, version 2
• oai:hal.archives-ouvertes.fr:hal-00535782
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• Submitted on: Tuesday, 20 March 2012 15:16:49
• Updated on: Thursday, 21 June 2012 19:27:12