Asymptotic analysis of the lattice Boltzmann equation

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier–Stokes equations, as opposed to the compressible Navier–Stokes equations obtained by the Chapman–Enskog analysis with convective scaling. We also apply the asymptotic analysis directly to the fully discrete LBE, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE. This leads to a consistency analysis which provides order-by-order information about the numerical solution of the LBE. The asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities, such as vorticity. It also justifies the use of Richardson’s extrapolation method. As an example, a two-dimensional Taylor-vortex flow is used to validate our analysis. The numerical results agree very well with our analytic predictions.     

Quasi-relativistic Boltzmann equation for systems with short-distance forces

Systems are referred to as quasirelativistic if terms up to the order of v²/c² suffice to describe them; v is the particle velocity, c is the light velocity. Systems of neutral particles are considered with nonvanishing interaction forces at such short distances that the interaction delay can be ignored. Equations are derived for the correlation functions using the Lagrange function which is known in the quasirelativistic approximation; hence using the N. N. Bogolyubov method the quasirelativistic analog is obtained of the Boltzmann equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 17, No. 2, pp. 78–82, February, 1974.In conclusion the authors consider it their pleasant duty to express their thanks to N. N. Bogolyubov, B. L. Bonch-Bruevich, and N. A. Chernikov for discussing with them the preprint of this article [9].     

Stability of the Boltzmann equation with potential forces on torus

In this paper, we are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in L¹-norm for any initial data with the finite total mass, mechanical energy and entropy.     

Inhomogeneous similarity solutions of the Boltzmann equation with confining external forces

The Nikolskii transform makes it possible to construct inhomogeneous solutions of the Boltzmann equation from homogeneous ones. These solutions correspond to a gas in expansion, but if we introduce external forces, they can relax toward absolute Maxwellians. This property holds independently of the assumed intermolecular inverse power force. Consequently, for Maxwell molecules and from energy-dependent homogeneous distributions, we construct effectively a class of inhomogeneous similarity distributions with Maxwellian equilibrium relaxation. We review and investigate again the homogeneous distributions which can be written in closed form, for instance, we show that an elliptic exact solution proposed some years ago violates positivity. For Maxwell interaction with singular cross sections, we numerically construct inhomogeneous distributions having Maxwellian equilibrium states and study the Tjon overshoot effect. We show that both the sign and the time decrease of the external force as well as the microscopic model of the cross section contribute to the asymptotic behavior of the distribution. These inhomogeneous similarity solutions include a class of distributions that asymptotically oscillate between different Maxwellians. Two classes of external forces are considered: linear spatial-dependent forces or linear velocity-dependent forces plus source term.     

Asymptotic behaviour of particle motion under repulsive forces

The asymptotic nature of motions, as time tends to infinity, is investigated for classical point particles interacting by repulsive two-body potentialsU _ij . It is found that the conditions are necessary and sufficient for asymptotically straight line uniform motion. In the case of equal asymptotic velocities the proof depends only on a certain property of the motion (partial center of mass convexity) implied by the repulsivity of the potentials.The original version of this paper was edited and re-written by A. Lenard from Indiana University. The author expresses his gratitude to Professor Lenard for his work which has significantly improved both the mathematical presentation and the style of the paper     

On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces

This paper considers the linear space-inhomogeneous Boltzmann equation in a convex, bounded or unbounded bodyD with general boundary conditions. First, mildL ¹-solutions are constructed in the cutoff case using monotone sequences of iterates in an exponential form. Assuming detailed balance relations, mass conservation and uniqueness are proved, together with anH-theorem with formulas for the interior and boundary terms. Local boundedness of higher moments is proved for soft and hard collision potentials, together with global boundedness for hard potentials in the case of a nonheating boundary, including specular reflections. Next, the transport equation with forces of infinite range is considered in an integral form. Existence of weakL ¹-solutions are proved by compactness, using theH-theorem from the cutoff case. Finally, anH-theorem is given also for the infinite-range case.     

Boltzmann equation with fluctuations

From the Liouville equation, by the method of multiple-time-scales, a generalized Boltzmann-equation with fluctuations is obtained on the statistical considerations of the randomness of the many-particle correlations in the macroscopic picture. These fluctuations lead to anH theorem in which theH function decreases, with fluctuations with time toward equilibrium. These fluctuations furnish a source for a random force term introduced by Fox and Uhlenbeck in the Boltzmann equation.     

Classical solution of the nonlinear Boltzmann equation in allR 3: Asymptotic behavior of solutions

Proof is given of the existence of a classical solution to the nonlinear Boltzmann equation in allR ³. The solution, which is global in time, exists if the initial data go to zero fast enough at infinity and the mean free path is sufficiently large. The solution is smooth in the space variable if the initial value is smooth. The asymptotic behavior of solutions is also given. It is shown that ast the solution to the Boltzmann equation can be approximated by the solution to the free motion problem.     

The Boltzmann equation: Global existence for a rare gas in an infinite vacuum

Solutions of the Boltzmann equation are proved to exist, globally in time, under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough. It is also proved, as might be expected in this case, that the density of the gas at each point in space goes to zero as time goes to infinity.Research supported by the Natural Science and Engineering Research Council Canada under Grant No. A-8560     

Application of the Boltzmann transport equation to calculations of flux and range distributions of energetic ions

An approximation method for calculating a flux and range distributions of energetic ions in a layer of matter is presented. Used successively for every layer of a medium, approximation equations describe the ion motion in the electronic stopping region.     

Iterative solution of the Boltzmann equation

We define an iterative scheme to solve the nonlinear Boltzmann equation. Conservation rules are maintained at each iterative step. We apply this method to a spatially uniform and isotropic velocity distribution function on the Maxwell and very-hard-particle models. A particular example is evaluated and results are compared with the exact solution. It shows to be a very fast convergent approach.     

Two generalizations of the Boltzmann equation

We connect two different extensions of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint part. Direct transformation formulas between key functions of the two approaches are given.     

Temperature transform of the Boltzmann equation

We define an integral transform of the energy distribution function for an isotropic and homogeneous diluted gas. It may be interpreted as a linear combination of equilibrium states with variable temperatures. We show that the temporal evolution features of the distribution function are determined by the singularities of this temperature transform. We compare the relaxation to the equilibrium process for Maxwell and very hard-particle interaction models, finding many basic discrepancies. Finally, we formulate an alternative approach, which is given by anN-pole approximation with a clear physical meaning.Fellow of the Conselho Nacional de Desenvolvimento Cientifico e Tecnólogico, Brazil.     

Nonisotropic solutions of the Boltzmann equation

We consider the relaxation to equilibrium of a spatially uniform Maxwellian gas. We expand the solution of the nonlinear Boltzmann equation in a truncated series of orthogonal functions. We integrate numerically the equation for non-isotropic initial conditions. For certain simple conditions we find interesting proximity effects and other transient relaxation phenomena at thermal energies. Furthermore, we define a resummation of the orthogonal expansion which is more convenient than the original one for the numerical analysis of the relaxation process.     

The Boltzmann equation with a soft potential

The results of Part I are extended to include linear spatially periodic problems-solutions of the initial value are shown to exist and decay like . Then the full non-linear Boltzmann equation with a soft potential is solved for initial data close to equilibrium. The non-linearity is treated as a perturbation of the linear problem, and the equation is solved by iteration.Supported by the National Science Foundation under Grant Nos. MCS78-09525 and MCS76-07039 and by the United States Army under Contract No. DAAG29-75-C-0024     

The Boltzmann equation with a soft potential

The initial value problem for the linearized spatially-homogeneous Boltzmann equation has the form ?f/?t+Lf=0 withf(ξ,t=0) given. The linear operatorL operates only on the ξ variable and is non-negative, but, for the soft potentials considered here, its continuous spectrum extends to the origin. Thus one cannot expect exponential decay forf, but in this paper it is shown thatf decays likee ^?λ t ^β with β<1. This result will be used in Part II to show existence of solutions of the initial value problem for the full nonlinear, spatially dependent problem for initial data that is close to equilibrium.     

A study on the inclusion of body forces in the lattice Boltzmann BGK equation to recover steady-state hydrodynamics

When the lattice Boltzmann (LB) method is used to solve hydrodynamic problems containing a body force term varying in space and/or time, its modelling at the mesoscopic scale must be verified in terms of consistency in order to avoid the appearance of non-hydrodynamic error terms at the macroscopic scale. In the present work it is shown that the modelling of spatially varying steady body force terms in the LB equation must be different from the time-dependent case, when a steady-state flow solution is sought. For that, the Chapman-Enskog analysis is used to derive the LB body force model for the LB BGK equations in a steady-state flow problem. The theoretical findings are supported by numerical tests performed on two different 2D steady-state laminar flows driven by spatially varying body forces with known analytical solutions.