Self-consistent Force Scheme in the Discrete Boltzmann Equation

Physica A: Statistical Mechanics and its Applications, 1992

This investigation presents a dynamical formulation of Boltzmann-like discrete kinetic equations. The central concept of such a formulation is based on the idea that a dissipative system is equipped with a functional Poisson bracket and a dissipative bracket, and with two functionals generating the dynamical evolution. We present non-canonical brackets and the generating functionals. We also discuss their basic properties and investigate dynamical invariants as well as the production of entropy generated by the entropy functional via the dissipative part of the bracket. When collisions are the only sources of dissipation we are coming up with a consistent up to now unknown version of the discrete Boltzmann-Vlasov kinetic equation.

arXiv: Fluid Dynamics, 2017

In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implementation of boundary conditions at the kinetic level. Such boundary conditions induce a discontinuity in the distribution function with respect to the component of the momentum which is normal to the boundary. Expanding the distribution function with respect to half-range polynomials allows this discontinuity to be captured. The implementation of this concept has been reported in the literature only for force-free flows. In the case of general forces which can have non-zero components in the direction perpendicular to the walls, the implementation of the force term requires taking the momentum space gradient of a discontinuous function. Our proposed method deals with this difficulty by employing the theory of distributions. We validate our procedure by considering the simple one-dimensional flow between diffuse-reflective walls of equal or different temperatures driven by the constant grav...

Transactions of the American Mathematical Society, 1972

In this paper we compare the nonlinear Boltzmann equation appearing in the kinetic theory of gases, with its linearized version. We exhibit an intertwining operator for the two semigroups involved. We do not assume from the reader any familiarity with Boltzmann's equation but rather start from scratch. 0. Introduction. Consider a dilute gas composed of a very large number of molecules moving in space according to the laws of classical mechanics, and colliding in pairs from time to time. Assume that we can disregard all external effects, such as gravity, so that the motion is completely specified by giving the intermolecular forces. One is interested in the number of molecules which at time t have position r and velocity v, within dr dv. This is given by n(t,r,v) = Nf(t,r,v)drdv(2) where/is called the density function. It is clear that this quantity is going to change in time due to the motion of the molecules and to the effect of the collisions. Boltzmann derived an equation for the rate of change off with time. It has the form of a nonlinear integro-differential equation: if' \f(vV)f(^-f(vi)f(v2)]\v,-v2\I(\v,-v2\, 8) sin 0dBd</>dv2.

Physical Review E, 1997

The lattice Boltzmann equation ͑LBE͒ is directly derived from the Boltzmann equation by discretization in both time and phase space. A procedure to systematically derive discrete velocity models is presented. A LBE algorithm with arbitrary mesh grids is proposed and a numerical simulation of the backward-facing step is conducted. The numerical result agrees well with experimental and previous numerical results. Various improvements on the LBE models are discussed, and an explanation of the instability of the existing LBE thermal models is also provided. ͓S1063-651X͑97͒51106-8͔

Il Nuovo Cimento B, 1974

Starting from the Liouville equation and making use of projection operator techniques we obtain ~ compact equation for the rate of change of the n-particle momentum distribution function to any order in thc density. This cquution is exact in the thermodynamic limit. The terms up to second order in the density ace studied, and expressions are given for the errors committed when one makes the usual hypothesis to derivc generalized Boltzmann equations. Finally the Choh-Uhlenbcck operator is obtained under additional assumptions. 1.-Introduction.

2005

Communications in Computational Physics

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number [17, 27, 30]. Unfortunately, however, such schemes are not conservative. Lack of conservation is analyzed in detail, and two main sources are identified as its cause. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grid points in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced in [22]. The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in [26]. In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

We study the formal precision of the relative velocity lattice Boltzmann schemes. They differ from the d&#39;Humi\`eres schemes by their relaxation phase: it occurs for a set of moments parametrized by a velocity field function of space and time. We deal with the asymptotics of the relative velocity schemes for one conservation law: the third order equivalent equation is exposed for an arbitrary number of dimensions and velocities.

Computer Physics Communications, 2000

Some rigorous results on discrete velocity models are briefly reviewed and their ramifications for the lattice Boltzmann equation (LBE) are discussed. In particular, issues related to thermodynamics and H -theorem of the lattice Boltzmann equation are addressed. It is argued that for the lattice Boltzmann equation satisfying the correct hydrodynamic equations, there cannot exist an H -theorem. Nevertheless, the equilibrium distribution function of the lattice Boltzmann equation can closely approximate the genuine equilibrium which minimizes the H -function of the corresponding continuous Boltzmann equation. It is also pointed out that the "equilibrium" in the LBE models is an attractor rather than a true equilibrium in the rigorous sense of H -theorem. Since there is no H -theorem to guarantee the stability of the LBE models at the attractor, the stability of the attractor can only be studied by means other than proving an H -function.

Physica D: Nonlinear Phenomena, 1993

Revista de la Unión Matemática …, 2007

We prove a theorem of existence, uniqueness and positivity of the solution for the Boltzmann equation with force term and initial data near the Vacuum.

2006

2003

The Lattice Boltzmann equations are usually constructed to satisfy physical requirements like Galilean invariance and isotropy as well as to possess a velocity-independent pressure, no compressible effects, just to mention a few. In this paper, a stability criterion for such constructions is introduced and is used to derive a new relation of the parameters in a parametrized 2-dimensional 9-velocity model.

The derivation,of the quantum,lattice Boltzmann,model,is reviewed,with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation,and the application to the computation,of the ground state of the Gross-Pitaevskii equation,(GPE). Numerical,results for the linear and non- linear Schr¨odinger,equation,and,for the ground,state solution of the GPE are also presented,and validated,against analytical results or other classical schemes,such as Crank-Nicholson. PACS: 02.70.-c, 03.65-w, 03.67.Lx Key words: Quantum lattice Boltzmann, multi-dimensions, imaginary-time model, linear and

1994

The problem of thermodynamic parameterization of an arbitrary approximation of reduced description is solved. On the base of this solution a new class of model kinetic equations is constructed that gives a model extension of the chosen approximation to a kinetic model. Model equations describe two processes: rapid relaxation to the chosen approximation along the planes of rapid motions, and the slow motion caused by the chosen approximation. The H-theorem is proved for these models. It is shown, that the rapid process always leads to entropy growth, and also a neighborhood of the approximation is determined inside which the slow process satisfies the H-theorem. Kinetic models for Grad moment approximations and for the Tamm-Mott-Smith approximation are constructed explicitly. In particular, the problem of concordance of the ES-model with the H-theorem is solved.