Numerical simulations of the Euler system with congestion constraint

In this paper, we study the numerical simulations for Euler system with maximal density constraint. This model is developed in [1, 3] with the constraint introduced into the system by a singular pressure law, which causes the transition of different asymptotic dynamics between different regions. To overcome these difficulties, we adapt and implement two asymptotic preserving (AP) schemes originally designed for low Mach number limit [2,4] to our model. These schemes work for the different dynamics and capture the transitions well. Several numerical tests both in one dimensional and two dimensional cases are carried out for our schemes

Resummed Kinetic Field Theory: Using Mesoscopic Particle Hydrodynamics to Describe Baryonic Matter in a Cosmological Framework

Recently, Bartelmann et al. presented a 'Kinetic Field Theory' (KFT) formalism to tackle the difficulties of large scale structure formation. In this approach, the dynamics of a non-equilibrium ensemble of classical particles are examined based on methods of statistical field theory. So far, only contributions coming from dark matter were considered, which is assumed to pose an accurate description of our universe on very large scales. Nevertheless, going to smaller scales, also baryonic contributions have to be taken into account. Building on the ideas of Viermann et al. we present an effective particle model of hydrodynamics to describe baryonic matter in a cosmological framework. Using this model, the baryonic density contrast power spectrum is computed to lowest perturbative order within the resummed KFT framework of Lilow et al. We discuss the qualitative differences to the dark matter case and perform a quantitative comparison to the baryonic spectrum obtained from Eulerian perturbation theory. A subsequent paper will resolve the problem of coupling both theories describing dark and baryonic matter, respectively, to gain a full model of cosmic matter. Even though our focus is on cosmological systems only, we want to emphasize that all methods presented here are of a quite general fashion, making it applicable also to other fields.Comment: 24 pages, 2 figures, current version: added more explanatory material (especially on the underlying RKFT-formalism), added references to literature on non-linear structure formation, make difference to pure dark matter model clearer, further minor changes; content matches published versio

Phase appearance or disappearance in two-phase flows

This paper is devoted to the treatment of specific numerical problems which appear when phase appearance or disappearance occurs in models of two-phase flows. Such models have crucial importance in many industrial areas such as nuclear power plant safety studies. In this paper, two outstanding problems are identified: first, the loss of hyperbolicity of the system when a phase appears or disappears and second, the lack of positivity of standard shock capturing schemes such as the Roe scheme. After an asymptotic study of the model, this paper proposes accurate and robust numerical methods adapted to the simulation of phase appearance or disappearance. Polynomial solvers are developed to avoid the use of eigenvectors which are needed in usual shock capturing schemes, and a method based on an adaptive numerical diffusion is designed to treat the positivity problems. An alternate method, based on the use of the hyperbolic tangent function instead of a polynomial, is also considered. Numerical results are presented which demonstrate the efficiency of the proposed solutions

Beyond pressureless gas dynamics: Quadrature-based velocity moment models

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.Comment: Submitted to Communication in Mathematical Science