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THE BOLTZMANN EQUATION ON A TWO-DIMENSIONAL LATTICE THEORETICAL AND NUMERICAL RESULTS
Abstract
The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. This paper contains a brief descrip- tion of the proof that was recently presented elsewhere ((L. Fainsilber, P. Kurlberg, B. Wennberg, preprint 2004)). It also presents the results of numerical experi- ments.
The construction of discrete velocity models or numerical methods for the
Boltzmann equation, may lead to the necessity of computing the collision
operator as a sum over lattice points. The collision operator involves an
integral over a sphere, which corresponds to the conservation of energy and
momentum. In dimension two there are difficulties even in proving the
convergence of such an approximation since many circles contain very few
lattice points, and some circles contain many badly distributed lattice points.
However, by showing that lattice points on most circles are equidistributed we
find that the collision operator can indeed be approximated as a sum over
lattice points in the two-dimensional case. For higher dimensions, this result
has already been obtained by A. Bobylev et. al. (SIAM J. Numerical Analysis 34
no 5 p. 1865-1883 (1997))
A numerical method for solving the Boltzmann equation is proposed. This method is based on a finite difference scheme for the approximation of the collision kernel and a finite element scheme for the transport phase. The properties of the Boltzmann equation are also satisfied in our method.
In this paper, we study the link between a certain class of discrete-velocity models (DVMs) and the Boltzmann equation. Those models possess an infinite number of velocities laying on a regular grid of step h in R3. Our aim is to prove that it is possible to construct models consistent with the Boltzmann equation, i.e., such that the discrete collision term can be seen as an approximation of the collision integral of the Boltzmann equation.
We prove the convergence of finite-difference approximations to solutions of the Boltzmann equation. An essential step is the proof of convergence of discrete approximations to the collision integral. This proof relies on our previous results on the consistency of this approximation. For the space-homogeneous problem we prove strong convergence of our discrete approximation to the strong solution of the Boltzmann equation. In the space-dependent case we prove weak convergence to DiPerna–Lions solutions.
We present a result of consistency of a discrete velocity model with the Boltzmann equation. This model is constructed as a quadrature formula of the collision integral, by using a regular grid in the velocity space ℝ 3 . The convergence of the above approximation leans mainly on some property of uniform repartition of integer roots of the equation p 2 +q 2 +r 2 =n, where n is an integer.
We study some questions related to general discrete velocity (with arbitrarily number of velocities) models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed. These results are applied to typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as a limiting case of the corresponding discrete models. Existence of solutions of weakly non-linear half-space problems for general DVMs are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Both implicit, in the non-degenerate cases, and sometimes, in both degenerate and non-degenerate cases, explicit conditions are found. Shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians) for DVMs. We give a constructive proof for the existence of solutions of the shock-wave problem for the general DVM. This is worked out for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. We clarify how close the shock speed must be for our theorem to hold, and present an iteration scheme for obtaining the solution. The main results of the paper can be used for DVMs for mixtures as well as for DVMs for one species.
We study the distribution of lattice points a + 1b on the fixed circle a2 + b2 = n. Our results apply p.p. to the representable integers n, and we supply bounds for the discrepancy of the distribution, and for the maximum and minimum of the angles between consecutive points. As a corollary, we are able to show that when n is representable then it is almost surely representable with min(a, b) small, with an explicit bound.
In the case of mixtures only a couple of discrete velocity models which have spurious invariants, have been introduced. Here we thoroughly investigate the matter, and supply examples of models with both finitely and infinitely many velocities without spurious invariants.
Two convergence results related to the approximation of the Boltzmann equation by discrete velocity models are presented. First we construct a sequence of deterministic discrete velocity models and prove convergence (as the number of discrete velocities tends to infinity) of their solutions to the solution of a spatially homogeneous Boltzmann equation. Second we introduce a sequence of Markov jump processes (interpreted as random discrete velocity models) and prove convergence (as the intensity of jumps tends to infinity) of these processes to the solution of a deterministic discrete velocity model.
Models for mixtures of discrete velocity gases have been recently introduced by the authors [J. Stat. Phys. 91, 327-342 (1998; Zbl 0917.76075) and Discrete velocity models: the case of mixtures, Transp. Theory Stat. Phys. 29, 209-216 (2000)] and have produced unexpected results, particularly with regard to the possible existence of nonphysical collision invariants. Here we discuss a method to construct models without spurious invariants. The method can be extended to very general models, including polyatomic gases, chemical reactions, etc.
The construction of discrete velocity models or numerical methods for the
Boltzmann equation, may lead to the necessity of computing the collision
operator as a sum over lattice points. The collision operator involves an
integral over a sphere, which corresponds to the conservation of energy and
momentum. In dimension two there are difficulties even in proving the
convergence of such an approximation since many circles contain very few
lattice points, and some circles contain many badly distributed lattice points.
However, by showing that lattice points on most circles are equidistributed we
find that the collision operator can indeed be approximated as a sum over
lattice points in the two-dimensional case. For higher dimensions, this result
has already been obtained by A. Bobylev et. al. (SIAM J. Numerical Analysis 34
no 5 p. 1865-1883 (1997))
This book is devoted to the presentation of rigorous mathematical results in the kinetic theory of a gas of hard spheres. Recent developments as well as classical results are presented in a unified way, such that the book should become the standard reference on the subject. There is no such book available at present. The reader will find a systematic treatment of the main mathematical results, a discussion of open problems, and a guide to the existing literature. There is a rigorous and comprehensive presentation of strict validation of the Boltzmann equations, global existence theory, and the fluid-dynamical limits. The authors also review and discuss classical derivation and properties of the Boltzmann equation, particle simulation methods, and boundary conditions.
This paper discusses the convergence of a new discrete-velocity model to the Boltzmann equation. First the consistency of the collision integral approximation is proved. Based on this we prove the convergence of solutions for a modied model to DiPerna-Lions solutions of the Boltzmann equation. As a test numerical example, the solutions to the discrete problems are compared to the exact solution of the Boltzmann equation in the space homogeneous case.
For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space S(R ). The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.
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E-mail address: kurlberg@math.kth.se DEPARTMENT OF MATHEMATICS
- Sweden Gothenburg
GOTHENBURG, SWEDEN
E-mail address: laura@math.chalmers.se
DEPARTMENT OF MATHEMATICS, ROYAL INSTITUTE OF TECHNOLOGY, SE-10044 STOCK-HOLM, SWEDEN
E-mail address: kurlberg@math.kth.se
DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96