In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the diierential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit ...

By combining the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation and the Dudyński and Ekiel-Jeżewska device of the causality of the relativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic...

Some rigorous results on discrete velocity models are briefly reviewed and their ramifications for the lattice Boltzmann equation (LBE) are discussed. In particular, issues related to thermodynamics andH -theorem of the lattice Boltzmann equation are addressed. It is argued that for the lattice Boltzmann equation satisfying the correct hydrodynamic equations, there cannot exist an H -theorem. N...

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self tim...

Some rigorous results on discrete velocity models are briefly reviewed and their ramifications for the lattice Boltzmann equation (LBE) are discussed. In particular, issues related to thermodynamics and H-theorem of the lattice Boltzmann equation are addressed. It is argued that for the lattice Boltzmann equation satisfying the correct hydrodynamic equations, there cannot exist an H-theorem. Ne...

electrostatic potential as well as the local volume charge density are computed for a macromolecule by solving the poisson-boltzmann equation (pbe) using the finite element method (fem). as a verification, our numerical results for a one dimensional pbe, which corresponds to an infinite-length macromolecule, are compared with the existing analytical solution and good agreement is found. as a ma...