A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck (LFP) equations by Boltzmann equations of quasi-Maxwellian kind. High-frequency fields are included into consideration and comparison with the well-known results are given.

The analytically difficult problem of existence of shock wave solutions is studied for the general discrete velocity model (DVM) with an arbitrary finite number of velocities (the discrete Boltzmann equation in terminology of H. Cabannes). For the shock wave problem the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this work we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed , corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is then shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998 [1]. In their technical proof Bose et al. are following the lines of the pioneering work for the continuous Boltzmann equation by Caflisch and Nicolaenko [2]. In this work, we follow a more straightforward way, suiting the discrete case. Our approach is based on results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points [3] to general dynamical systems of the same type as in the shock wave problem for DVMs. Our proof is constructive, and it is also shown (at least implicitly) how close to the typical speed , the shock speed must be for our results to be valid. All results are mathematically rigorous. Our results are also applicable for DVMs for mixtures. ACKNOWLEDGEMENTS. The support by the Swedish Research Council grant 20035357 are gratefully acknowledged by both of the authors.REFERENCES[1] C. Bose, R. Illner, S. Ukai, Transp. Th. Stat. Phys., 27, 35-66 (1998) [2] R.E. Caflisch, B. Nicolaenko, Comm. Math. Phys., 86, 161-194 (1982)[3] A.V. Bobylev, N. Bernhoff, Lecture Notes on the Discretization of the Boltzmann Equation, World Scientific, 2003, pp. 203-222

We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed c, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive proof following a more straightforward way, suiting the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points to general dynamical systems of the same type as in the shock wave problem for DVMs. The same approach can also be applied for DVMs for mixtures

Equations of hydrodynamics (derived from the Boltzmann equation) beyond the Navier-Stokes level are studied by a method proposed earlier by the author. The main question we consider is the following: What is the most natural replacement for classical (ill-posed) Burnett equations?

It is shown that, in some sense, it is a two-parameter set of Generalized Burnett Equations (GBEs) derived in this paper. Some equations of this class are even simpler than original Burnett equations. The region of stability in the space of parameters and other properties of GBEs are discussed.

Stationary shock wave solutions for the generalized Burnett equations (GBE) [ A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys. 132, 569 (2008) ] are studied. Based on the results of Bisi et al. [Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models 1, 295 (2008) ], we choose a unique (optimal) form of GBE and solve numerically the shock wave problem for various Mach numbers. The results are compared with the numerical solutions of NavierStokes equations and with the MottSmith approximation for the Boltzmann equation (all calculations are done for Maxwell molecules) since it is believed that the MottSmith approximation yields better results for strong shocks. The comparison shows that GBE yield certain improvement of the NavierStokes results for moderate Mach numbers

Numerical solution of a fully nonlinear one dimensional in space and three dimensional in velocity space electron kinetic equation is presented. Direct Simulation Monte Carlo (DSMC) method used for the nonlinear Landau-Fokker-Planck (LFP) collision operator is combined with Particle-in-Cell (PiC) simulations. An assumption of a self-consistent ambipolar electric field is used. The illustrative simulation results for the relaxation of the initial temperature perturbation are compared with the antecedent analytical and numerical results.

Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economics, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. A new application of multi-linear models to economics and social dynamics is discussed

 In this paper we consider Lie group symmetries of evolutionequations with non-local operators in context of applications tononlinear kinetic equations. As an illustration we consider theBoltzmann equation and calculate the admitted group of pointtransformations.               

We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the 2-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.

It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta-function) can be presented as a classof solutions of the Fourier transformed spatially homogeneous linear Boltzmannequation with a special Maxwell-type collision kernel. The result is based on anexplicit integral representationof solutions to the Cauchy problem for the Boltzmann equation. Possibleapplications to the theory of Dirichlet seriesare briefly discussed.

The new direct simulation method of Monte-Carlo type (DSMC) for Coulomb collisions in the case of two component plasma is considered.  A brief literature review and preliminary information concerning the problem are given. Further the idea that lies in the basis of the method is discussed and its scheme is provided. The illustrative numerical simulation of the initial distribution relaxation for one and two sorts of particles in 3D case in the velocity space is performed. Simulation results are compared with the numerical results based on the completely conservative finite difference schemes for the Landau-Fokker-Planck equation. Estimation of calculation accuracy obtained from numerical results is given.

In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the clas-sical Burnett equations these equations are well-posed and therefore can be used in applications. We considerthe one-dimensional version of the generalized Burnett equations for Maxwell molecules in both Eulerian andLagrangian coordinates and perform the complete group analysis of these equations. In particular, this includesfinding and analyzing admitted Lie groups. Our classifications of the Lie symmetries of the Navier-Stokes equa-tions of compressible gas and generalized Burnett equations provide a basis for finding invariant solutions ofthese equations. We also consider representations of all invariant solutions. Some particular classes of invariantsolutions are studied in more detail by both analytical and numerical methods

A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck equations by Boltzmann equations of quasi-Maxwellian kind. It means that the total collision frequency for the corresponding Boltzmann equation does not depend on the velocities. This allows to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes the well-known methods of Takizuka and Abe (1977) [12] and Nanbu (1997) as particular cases, and generalizes the approach of Bobylev and Nanbu (2000). The numerical scheme of this paper is simpler than the schemes by Takizuka and Abe [12] and by Nanbu. We derive it for the general case of multicomponent plasmas and show some numerical tests for the two-component (electrons and ions) case. An optimal choice of parameters for speeding up the computations is also discussed. It is also proved that the order of approximation is not worse than O(root epsilon), where epsilon is a parameter of approximation being equivalent to the time step Delta t in earlier methods. A similar estimate is obtained for the methods of Takizuka and Abe and Nanbu. (C) 2013 Elsevier Inc. All rights reserved.

A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of the Landau-Fokker-Planck equations by the Boltzmann equations of a quasi-Maxwellian kind. This means that the total collision frequency for the corresponding Boltzmann equation does not depend on velocities. This allows one to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes (as particular cases) the well-known methods of Takizuka & Abe(1977) and Nanbu(1997) and generalizes the approach of Bobylev & Nanbu(2000). The numerical scheme of this paper is simpler than the schemes by Takizuka & Abe and by Nanbu. We derive it for the general case of multicomponent plasmas

We consider a system of N classical particles, interacting via a smooth, short-range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey to the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in term of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N →∞, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.

The linear and the linearized Boltzmann collision operators are studied in the case of hard spheres. The equations for the integral operators are reduced to ordinary differential equations. From these equations, we show that the collision operators have discrete eigenvalues, and we demonstrate how to compute them. We also use the differential equations to investigate asymptotics for the linearized collision operator.

In this paper we discuss a general problem related to spurious conservation laws for discrete velocity models (DVMs) of the classical (elastic) Boltzmann equation. Models with spurious conservation laws appeared already at the early stage of the development of discrete kinetic theory. The well-known theorem of uniqueness of collision invariants for the continuous velocity space very often does not hold for a set of discrete velocities. In our previous works we considered the general problem of the construction of normal DVMs, we found a general algorithm for the construction of all such models and presented a complete classification of normal DVMs with small number n of velocities (n<11). Even if we have a general method to classify all normal discrete kinetic models (and in particular DVMs), the existing method is relatively slow and the amount of possible cases to check increases rapidly with n. We remarked that many of our normal DVMs appear to be axially symmetric. In this paper we consider a connection between symmetric transformations and normal DVMs. We first develop a new inductive method that, starting with a given normal DVM, leads by symmetric extensions to a new normal DVM. This method can produce very fast many new normal DVMs with larger number of velocities, showing that the class of normal DVMs contains a large subclass of symmetric models. We finally apply the method to several normal DVMs and construct new models that are not only normal, but also symmetric relatively to more and more axes. We hope that such symmetric velocitysets can be used for DSMC methods of solving Boltzmann equation.

In the present work we consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by R. Gatignol [1] in connection with discrete models of the Boltzmann equation (BE), when it became clear that the velocity discretization can lead to equations with spurious conservation laws (not linear combinations of physical invariants). The problem has been addressed in the last decade by several authors, in particular by Cercignani, Bobylev, Vedenyapin, and Cornille. Even though a practical criterion for the non-existence of spurious conservation laws has been devised [2], and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed [3], a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We develop such a general algorithm in the present work.

We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d 2 and for any sufficiently large number N of velocities (for example, N 6 for the planar case d = 2) there exists just a finite number of distinct (non-equivalent) classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. In the first case, we show that all normal models can be explicitly described. In the second case, we give a complete classification of normal models up to 9 velocities and show that the extension method [3], does not lead to all normal DVMs.

Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of one-kind particles of the single gases involved in the mixture, the corresponding discrete models for a single gas are also normal models). We apply this method and obtain SNMs with up to 20 velocities and their spectrum of mass ratio.

Finally, we develop a new method that can lead, by symmetric transformations, from a given normal DVM to extended normal DVMs.



ACKNOWLEDGEMENTS. The support by the grant 2003-5357 from Swedish Research Council for both authors is gratefully acknowledged.



REFERENCES

[1] R. Gatignol, Théorie Cinétique des Gaz à Répartition Discrète de Vitesses, Springer-Verlag, New-York, 1975

[2] V. V. Vedenyapin, Y. N. Orlov, Teoret. and Math. Phys., 121, 1516-1523 (1999)

[3] A. V. Bobylev, C. Cercignani, J. Statist. Phys., 97, 677-686 (1999)

The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.

                 We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters where is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.