Stability and convergence of discrete kinetic approximations to an initial-boundary value problem for conservation laws

We present some new convergence results for a discrete velocities BGK approximation to an initial boundary value problem for a single hyperbolic conservation law. In this paper we show stability and convergence toward a unique entropy solution in the general framework without any restriction either on the data of the limit problem or on the set of velocity of the BGK model.

     

A BGK Model for Gas Mixtures

The aim of this article is to construct a BGK operator for gas mixtures starting from the true Navier-Stokes equations. That is the ones having transport coefficients given by the hydrodynamical limit of the Boltzmann equation(s). Here the same hydrodynamical limit is obtained by introducing relaxation coefficients on certain moments of the distribution functions. Next the whole model is set by using entropy minimization under moments constraints. In our case the BGK operator allows to recover the exact Fick and Newton laws and satisfy the fundamental properties of the Boltzmann equations for inert gas mixtures.     

Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles

Dissipative solutions [12] of the Euler equations of incompressible fluids are obtained as the hydrodynamic limit of a properly scaled BGK equation. This stability result comes from refined entropy and entropy dissipation bounds. It uses in a crucial way the local conservation laws which are known to hold for weak solutions of this simplified model of the Boltzmann equation.     

Variational approach to homoclinic orbits in twist maps and an application to billiard systems

We study in this article the topological entropy of billiard systems on a convex domain of the Euclidean plane. We restrict our attention to those systems whose boundary curve has positive curvature and show that for generic billiard ball systems satisfying this condition the topological entropy is positive.     

Global weak solutions to the relativistic BGK equation

Abstract

In this paper, the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity, and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.     

Stability and consistency of kinetic upwinding for advection-diffusion equations

^** Email: matorril{at}ust.hk^*** Email: makxu{at}ust.hk Numerical methods based on kinetic models of fluid flows, likethe so-called BGK scheme, are becoming increasingly popularfor the solution of convection-dominated viscous fluid equationsin a finite-volume approach due to their accuracy and robustness.Based on kinetic-gas theory, the BGK scheme approxi-mately solvesthe BGK kinetic model of the Boltzmann equation at each cellinterface and obtains a numerical flux from integration of thedistribution function. This paper provides the first analyticalinvestigations of the BGK-scheme and its stability and consistencyapplied to a linear advection–diffusion equation. Thestructure of the method and its limiting cases are discussed.The stability results concern explicit time marching and demonstratethe upwinding ability of the kinetic method. Furthermore, itsstability domain is larger than that of common finite-volumemethods in the under-resolved case, i.e. where the grid Reynoldsnumber is large. In this regime, the BGK scheme is shown toallow the time step to be controlled from the advection alone.We show the existence of a third-order ‘super-convergence’on coarse grids independent of the initial condition. We alsoprove a limiting order for the local consist-ency error andshow the error of the BGK scheme to be asymptotically firstorder on very fine grids. However, in advection-dominated regimessuper-convergence is responsible for the high accuracy of themethod.     

A discrete BGK approximation for strongly degenerate parabolic problems with boundary conditions

We consider a class of BGK systems with a finite number of velocities, depending on a positive relaxation parameter, that approximate strongly degenerate hyperbolic-parabolic equations with initial boundary conditions. We prove a priori estimates for the solutions of the systems, showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero.     

Convergence d'un modèle à vitesses discrètes pour l'équation de Boltzmann-BGK

We consider a conservative and entropie discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. First, we prove a consistency result for this approximation. Then, we demonstrate that the discrete-velocity model possesses a unique solution. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.     

Non-extinction of a Fleming-Viot particle model

We consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.     

Stationary measures and hydrodynamics of zero range processes with several species of particles

We study general zero range processes with different types of particles on a d-dimensional lattice with periodic boundary conditions. A necessary and sufficient condition on the jump rates for the existence of stationary product measures is established. For translation invariant jump rates we prove the hydrodynamic limit on the Euler scale using Yaus relative entropy method. The limit equation is a system of conservation laws, which is hyperbolic and has a globally convex entropy. We analyze this system in terms of entropy variables. In addition we obtain stationary density profiles for open boundaries.     

Approximation of Solutions of the Neumann Problem in Disintegrating Domains

This article deals with approximation of solutions of the Neumann problem in domains, where small tubes are cut out. With an increasing number of tubes some kind of a porous layer inside the domain is approximated. Our aim is to find an asymptotic solution for the separated limit domains. We show that this asymptotics is described by a boundary value problem for the two limit domains, where the solutions for each domain are connected by the boundary conditions.     

局部熵高维重分形谱的上界估计

We discuss the problem of higher-dimensional multifractal spectrum of local entropy for arbitrary invariant measures. By utilizing characteristics of a dynamical system, namely, higher-dimensional entropy capacities and higher-dimensional correlation entropies, we obtain three upper estimates on the hlgher-dimensional multifractal spectrum of local entropies. We also study the domain of higher-dimensional multifractai spetrum of entropies.     

Un modèle ES–BGK pour des mélanges de gaz

In this note, we derive an ES–BGK model for gas mixtures that is able to give the correct Prandtl number obtained from the true Boltzmann equation. The derivation principle is based on the resolution of an entropy minimisation problem under moments constraints. The set of constraints is constructed by introducing a relaxation of some non-conserved moments. The non-conserved moments are dissipated according to some relaxation rates. Finally the BGK model that is obtained satisfies the classical properties of the Boltzmann operator for inert gas mixtures.     

Weak solutions for a hyperbolic system with unilateral constraint and mass loss

We consider isentropic gas dynamics equations with unilateral constraint on the density and mass loss. The γ and pressureless pressure laws are considered. We propose an entropy weak formulation of the system that incorporates the constraint and Lagrange multiplier, for which we prove weak stability and existence of solutions. The nonzero pressure model is approximated by a kinetic BGK relaxation model, while the pressureless model is approximated by a sticky-blocks dynamics with mass loss.     

Viscosity method in a transonic flow

We prove that the solution of a nonviscous compressible transonic flow can be obtained as a limit of viscous solutions, if the viscosity and heat conductivity tend to zero. To obtain an isentropic irrotational flow it is necessary to control the entropy and temperature on the boundary in a convenient way     

Entropy Solutions to a Second Order Forward-Backward Parabolic Differential Equation

We prove that the first boundary value problem for a second order forward-backward parabolic differential equation in a bounded domain G _T ^d+1, where d 2, has a unique entropy solution in the sense of F. Otto. Under some natural restrictions on the boundary values this solution is constructed as the limit with respect to a small parameter of a sequence of solutions to Dirichlet problems for an elliptic differential equation. We also show that the entropy solution is stable in the metric of L ₁(G _T ) with respect to perturbations of the boundary values in the metric of L ₁(G _T ).Original Russian Text Copyright © 2005 Kuznetsov I. V.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00829).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 594–619, May–June, 2005.     

On iterative and Monte Carlo solutions of the Boltzmann equation

The two most commonly used techniques for solving the Boltzmann equation, with given boundary conditions, are first iterative equations (typically the BGK equation) and Monte Carlo methods. The present work examines the accuracy of two different iterative solutions compared with that of an advanced Monte Carlo solution for a one-dimensional shock wave in a hard sphere gas. It is found that by comparison with the Monte Carlo solution the BGK model is not as satisfactory as the other first iterative solution (Holway's) and that the BGK solution may be improved by using directional temperatures rather than a mean temperature.     

Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains

We establish the existence of the classical solution for the pressure-gradient equation in a non-smooth and non-convex domain. The equation is elliptic inside the domain, becomes degenerate on the boundary, and is singular at the origin when the origin lies on the boundary. We show the solution is smooth inside the domain and continuous up to the boundary.     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.