On the sizes of discrete velocity models of the Boltzmann equation for mixtures

The Boltzmann equation for a mixture of particles with different masses is modeled using symmetric discrete velocity models that involve energy interchange between the species of the mixture. The computational complexity of this problem is investigated. New discrete models are presented. Published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 6, pp. 1045–1054.     

On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas

The Broadwell model of the Boltzmann equation for a simple discrete velocity gas is investigated on two asymptotic problems. (a) The decay of solutions inxR ast+. (b) The hydrodynamical limit in the compressible Euler level as the mean free path0.     

Conservation laws for polynomial Hamiltonians and for discrete models of the Boltzmann equation

Conservation laws that are linear with respect to the number of particles are constructed for classical and quantum Hamiltonians. A class of relaxation models generalizing discrete models of the Boltzmann equation are also considered. Conservation laws are written for these models in the same form as for the Hamiltonians. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 307–315, November, 1999.     

Generalized Broadwell models for the discrete Boltzmann equation with linear and quadratic terms

A generalization of the Broadwell models for the discrete Boltzmann equation with linear and quadratic terms is investigated. We prove that there exists a time‐global solution to this model in one space‐dimension for locally bounded initial data, using a maximum principle of solutions. The boundedness of solutions is established by analyzing the system of ordinary equations related to the linear term. Copyright © 2000 John Wiley & Sons, Ltd.     

Survey of initial value problem for the discrete Boltzmann equation

This paper is devoted to the story of the initial value problem in discrete kinetic theory. Starting from the first global existence theorem obtained by Nishida and Mimura in 1974 for the Broadwell model, which is the simplest discrete model of the Boltzmann equation, we continue with the extension of the results to the cases of more complex models, in particular models with multiple collisions; we notify works in which are considered cases with more than one space variable, and also problems in which the initial data can be partially negative. Conferenza tenuta il 29 settembre 1992     

Global existence for two-velocity models of the Boltzmann equation

If A is a symmetric 2 × 2-matrix, then the initial value problem describes the evolution in time of a fictive gas whose particles can move only with the velocities u₁ and v₂. It is proved that, for continuous initial values vanishing at infinity, (1) has a global solution if an H-Theorem holds for the gas described by (1). The validity of an H-Theorem is expressed by the properties of A.     

On Fatou's Lemma in several dimensions

Probability Theory and Related Fields -     

On the asymptotic behaviour and stability of the solution for the Broadwell model of the Boltzmann equation in three dimensions

The initial value problem for the full Broadwell model, in gas kinetic theory, is investigated. By means of a fixed point theorem an upper estimate for the solution is derived, provided that the initial values satisfy suitable a priori conditions. Since this estimate is function of the time, the asymptotic behaviour of the global solution is determined.     

A new consistent discrete‐velocity model for the Boltzmann equation

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 modified model to renormalized solutions of the Boltzmann equation. In a numerical example, the solutions to the discrete problems are compared with the exact solution of the Boltzmann equation in the space‐homogeneous case. Copyright © 2002 John Wiley & Sons, Ltd.     

On a simulation scheme for the Boltzmann equation

A scheme for the simulation of solutions of the Boltzmann equation derived by Nanbu is investigated. Rigorous results concerning questions of justification, the computation effort and the energy fluctuations are presented.     

On fluid limit for the semiconductors Boltzmann equation

This paper is devoted to the derivation of (non-linear) drift-diffusion equations from the semiconductor Boltzmann equation. Collisions are taken into account through the non-linear Pauli operator, but we do not assume relation on the cross section such as the so-called detailed balance principle. In turn, equilibrium states are implicitly defined. This article follows and completes the contribution of Mellet (Monatsh. Math. 134 (4) (2002) 305-329) where the electric field is given and does not depend on time. Here, we treat the self-consistent problem, the electric potential satisfying the Poisson equation. By means of a Hilbert expansion, we shall formally derive the asymptotic model in the general case. We shall then rigorously prove the convergence in the one-dimensional case by using a modified Hilbert expansion.     

Scattering of small solutions to generalized benjamin-bona-mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation u_t-△u_t=(b,▽u)+u^p(a,▽u)in x?Rⁿ,n≥2, with integer p≥3 , decays to zero like t^-2/3 if n=2 and like t^-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation u_t-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.     

On two step Lax-Wendroff methods in several dimensions

A version of Richtmyer's two step Lax-Wendroff scheme for solving hyperbolic systems in conservation form, is considered. This version uses only the nearest points, has second order accuracy at every time cycle and allows a time step which is larger by a factor of than Richtmyer's, whered is the number of spatial dimensions. The scheme appears to be competitive with the optimal stability schemes proposed by Strang and carried out by Gourlay and Morris.     

On the solutions of generalized discrete Poisson equation

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a constant. The proof uses the Fourier transform as the main tool. The necessary condition for the existence of the solution is provided.     

On iterative and Monte Carlo solutions of the Boltzmann equation

The two most commonly used techniques for solving the Boltzmann equation, with given boundary conditions, are first iterative equations (typically the BGK equation) and Monte Carlo methods. The present work examines the accuracy of two different iterative solutions compared with that of an advanced Monte Carlo solution for a one-dimensional shock wave in a hard sphere gas. It is found that by comparison with the Monte Carlo solution the BGK model is not as satisfactory as the other first iterative solution (Holway's) and that the BGK solution may be improved by using directional temperatures rather than a mean temperature.     

On the existence of non-negative numerical solutions of the Boltzmann equation

The diamond difference scheme approximating the linear Boltzmann equation may provide partly negative solutions. From the physics' point of view the solutions describing the density of neutrons or photons should be non-negative. It is shown that under assumptions being satisfied by suitable physical problems the solutions are non-negative if the step size is sufficiently small. This is shown for inhomogeneous boundary problems and for eigenvalue problems of the one-dimensional Boltzmann equation. In the latter case the greatest eigenvalue is a measure of the reactivity of reactors. It is proved that this eigenvalue is real and positive.