The work is devoted to some mathematical problems of dynamics of collisional plasma. The difficulty is that in plasma case we have at least three different length scales: Debye radius $$ r_D $$ , mean free path l and macroscopic length L. This is true even for the simplest model (plasma of electrons with a neutralizing background of infinitely heavy ions), considered in the paper. We study (at the formal level of mathematical rigour) solutions of the VLPE, having the typical length of the order $$ l>> r_D $$ , and try to clarify some mathematical questions related to corresponding limit. In particular, we study the existence of the limit for electric field and show that, generally speaking, it does not exist because of rapidly oscillating terms. An approximate asymptotic formula for the oscillating electric field near this limit is derived from VLPE. Still the limiting equations, which are used in many publications by physicists, can lead in some cases to correct results for the distribution function. Both these conclusions are confirmed by more explicit analysis of the linearized Vlasov–Poisson equation. We also study the well-posedness of limiting kinetic equations and the corresponding criterion in the class of weakly inhomogeneous initial data. It is shown that the collisional effects do not play an important role in this problem. In particular the equations are well-posed in the case of small deviations from equilibrium, as it was already known for related collisionless models.

We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and H- theorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

The Landau (Fokker–Planck) integral of Coulomb collisions is an integral component of the physical and mathematical models of both laboratory and space plasma, in which the intermediate collisionality regime is important. This paper discusses the method for the direct statistical simulation of the Monte Carlo type for a kinetic equation with a nonlinear operator of the Landau–Fokker–Planck (LFP) Coulomb collisions. This method is based on the approximation of the Landau collision integral by the Boltzmann collision integral. The paper has two main objectives: firstly, to obtain numerical estimates of the order of approximation of the Landau collision integral by the Boltzmann integral and, secondly, to explore the possibilities of optimizing the algorithm for multiply charged ions. The results are illustrated by the calculations of the problem on the relaxation of the initial distribution to equilibrium for one and two components.

This work deals with the numerical resolution of the M 1 -Maxwell system in the quasi-neutral regime. In this regime the stiffness of the stability constraints of classical schemes causes huge calculation times. That is why we introduce a new stable numerical scheme consistent with the transitional and limit models. Such schemes are called Asymptotic-Preserving (AP) schemes in literature. This new scheme is able to handle the quasi-neutrality limit regime without any restrictions on time and space steps. This approach can be easily applied to angular moment models by using a moments extraction. Finally, two physically relevant numerical test cases are presented for the Asymptotic-Preserving scheme in different regimes. The first one corresponds to a regime where electromagnetic effects are predominant. The second one on the contrary shows the efficiency of the Asymptotic-Preserving scheme in the quasi-neutral regime. In the latter case the illustrative simulations are compared with kinetic and hydrodynamic numerical results.

A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.

In this paper we consider a spatially homogeneous weakly collisional particle system, the state of which is described by a nonlinear kinetic equation with the Landau-Fokker-Planck collision integral and external heat sources. A quasi-linear diffusion operator and an external electric field are used as heating operators. Particle heating and acceleration and the runaway effect are analyzed. The numerical method used to study this phenomenon is the direct simulation Monte Carlo (DSMC) method.

272 KINETICS OF NONEQUILIBRIUM ELECTRON-PHONON SYSTEM FOR SEMICONDUCTORS AND METALS IN A STRONG ELECTRIC FIELD V.I. Karas ́, I.F. Potapenko, A.M. Vlasenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; M.V. Keldysh Institute of Applied Mathematics of the Russian Academy of Science, Moscow, Russian; V.N. Karazin Kharkov National University, Kharkov, Ukraine E-mail: karas@kipt.kharkov.ua; firena@yandex.ru; tn32@yandex.ru

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.

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(ε), where ε is a parameter of approximation being equivalent to the time step Δt in earlier methods. A similar estimate is obtained for the methods of Takizuka and Abe and Nanbu.