TL;DR: A new fast numerical method computing the asymptotic states and outgoing distributions for a linearized BGK half-space problem is presented and relations with the so-called variational methods are discussed.
Abstract: Linear half-space problems can be used to solve domain decomposition problems between Boltzmann and aerodynamic equations. A new fast numerical method computing the asymptotic states and outgoing distributions for a linearized BGK half-space problem is presented. Relations with the so-called variational methods are discussed. In particular, we stress the connection between these methods and Chapman-Enskog type expansions.
TL;DR: In this paper, an asymptotic-induced scheme for nonstationary transport equations with the diffusion scaling is developed, which works uniformly for all ranges of mean-free paths.
Abstract: An asymptotic-induced scheme for nonstationary transport equations with the diffusion scaling is developed. The scheme works uniformly for all ranges of mean-free paths. It is based on the asymptotic analysis of the diffusion limit of the transport equation.
A theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven. Moreover, numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically.
TL;DR: In this article, the micro-macro decomposition is extended to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics and two main improvements are presented: 1) a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics.
Abstract: In this work, we extend the micro-macro decomposition based numerical schemes developed in 
to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part.
A suitable discretization of this micro-macro model enables to derive an asymptotic preserving
scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented:
On the one hand a self-consistent electric field is introduced, which induces a specific discretization
in the velocity direction, and represents a wide range of applications in plasma physics.
On the other hand, as suggested in , we introduce a suitable reformulation of the micro-macro
scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit
scheme for the diffusion limit model when $\varepsilon\rightarrow 0$, which makes it free from the usual diffusion constraint
$\Delta t=O(\Delta x^2)$ in all regimes.
Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes.
Cites methods from "A Numerical Method for Computing As..."
...Finally, we need to determine the consistent boundary condition for the diffusion model; the exact condition is given by  but a good approximation is given by (see [16, 24, 25, 29])...
TL;DR: In this article, a numerical scheme for the nonstationary Boltzmann equation in the incompressible Navier-Stokes limit is developed, which works uniformly for all ranges of mean free paths.
Abstract: A numerical scheme for the nonstationary Boltzmann equation in the incompressible Navier--Stokes limit is developed. The scheme is induced by the asymptotic analysis of the Navier--Stokes limit for the Boltzmann equation. It works uniformly for all ranges of mean free paths. In the limit the scheme reduces to the Chorin projection method for the incompressible Navier--Stokes equation. Numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically.
TL;DR: An asymptotic-induced scheme for kinetic semiconductor equations with the diffusion scaling is developed and works uniformly for all ranges of mean free paths.
Abstract: An asymptotic-induced scheme for kinetic semiconductor equations with the diffusion scaling is developed. The scheme is based on the asymptotic analysis of the kinetic semiconductor equation. It works uniformly for all ranges of mean free paths. The velocity discretization is done using quadrature points equivalent to a moment expansion method. Numerical results for different physical situations are presented.
TL;DR: In this paper, the authors deal with domain decomposition methods for kinetic and drift diffusion semiconductor equations and give accurate coupling conditions at the interface between the kinetic and the drift diffusion domain.
Abstract: This paper deals with domain decomposition methods for kinetic and drift diffusion semiconductor equations. In particular accurate coupling conditions at the interface between the kinetic and drift diffusion domain are given. The cases of slight and strong nonequilibrium situations at the interface are considered and numerical examples are shown.
TL;DR: In this article, the Boltzmann Equation for rigid spheres is used to model the dynamics of a gas of rigid spheres in phase space and to solve the problem of flow and heat transfer in regions bounded by planes or cylinders.
Abstract: I. Basic Principles of The Kinetic Theory of Gases.- 1. Introduction.- 2. Probability.- 3. Phase space and Liouville's theorem.- 4. Hard spheres and rigid walls. Mean free path.- 5. Scattering of a volume element in phase space.- 6. Time averages, ergodic hypothesis and equilibrium states.- References.- II. The Boltzmann Equation.- 1. The problem of nonequilibrium states.- 2. Equations for the many particle distribution functions for a gas of rigid spheres.- 3. The Boltzmann equation for rigid spheres.- 4. Generalizations.- 5. Details of the collision term.- 6. Elementary properties of the collision operator. Collision invariants.- 7. Solution of the equation Q(f,f) = 0.- 8. Connection between the microscopic description and the macroscopic description of gas dynamics.- 9. Non-cutoff potentials and grazing collisions. Fokker-Planck equation.- 10. Model equations.- References.- III. Gas-Surface Interaction and the H-Theorem.- 1. Boundary conditions and the gas-surface interaction.- 2. Computation of scattering kernels.- 3. Reciprocity.- 4. A remarkable inequality.- 5. Maxwell's boundary conditions. Accommodation coefficients.- 6. Mathematical models for gas-surface interaction.- 7. Physical models for gas-surface interaction.- 8. Scattering of molecular beams.- 9. The H-theorem. Irreversibility.- 10. Equilibrium states and Maxwellian distributions.- References.- IV, Linear Transport.- 1. The linearized collision operator.- 2. The linearized Boltzmann equation.- 3. The linear Boltzmann equation. Neutron transport and radiative transfer.- 4. Uniqueness of the solution for initial and boundary value problems.- 5. Further investigation of the linearized collision term.- 6. The decay to equilibrium and the spectrum of the collision operator.- 7. Steady one-dimensional problems. Transport coefficients.- 8. The general case.- 9. Linearized kinetic models.- 10. The variational principle.- 11. Green's function.- 12. The integral equation approach.- References.- V. Small and Large Mean Free Paths.- 1. The Knudsen number.- 2. The Hilbert expansion.- 3. The Chapman-Enskog expansion.- 4. Criticism of the Chapman-Enskog method.- 5. Initial, boundary and shock layers.- 6. Further remarks on the Chapman-Enskog method and the computation of transport coefficients.- 7. Free molecule flow past a convex body.- 8. Free molecule flow in presence of nonconvex boundaries.- 9. Nearly free-molecule flows.- References.- VI. Analytical Solutions of Models.- 1. The method of elementary solutions.- 2. Splitting of a one-dimensional model equation.- 3. Elementary solutions of the simplest transport equation.- 4. Application of the general method to the Kramers and Milne problems.- 5. Application to the flow between parallel plates and the critical problem of a slab.- 6. Unsteady solutions of kinetic models with constant collision frequency.- 7. Analytical solutions of specific problems.- 8. More general models.- 9. Some special cases.- 10. Unsteady solutions of kinetic models with velocity dependent collision frequency.- 11. Analytic continuation.- 12. Sound propagation in monatomic gases.- 13. Two-dimensional and three-dimensional problems. Flow past solid bodies.- 14. Fluctuations and light scattering.- References.- VII. The Transition Regime.- 1. Introduction.- 2. Moment and discrete ordinate methods.- 3. The variational method.- 4. Monte Carlo methods.- 5. Problems of flow and heat transfer in regions bounded by planes or cylinders.- 6. Shock-wave structure.- 7. External flows.- 8. Expansion of a gas into a vacuum.- References.- VIII. Theorems on the Solutions of the Boltzmann Equation.- 1. Introduction.- 2. The space homogeneous case.- 3. Mollified and other modified versions of the Boltzmann equation.- 4. Nonstandard analysis approach to the Boltzmann equation.- 5. Local existence and validity of the Boltzmann equation.- 6. Global existence near equilibrium.- 7. Perturbations of vacuum.- 8. Homoenergetic solutions.- 9. Boundary value problems. The linearized and weakly nonlinear cases.- 10. Nonlinear boundary value problems.- 11. Concluding remarks.- References.- References.- Author Index.
TL;DR: In this paper, the authors present a survey of abstract kinetic theory and its application in various areas of physics, chemistry, biology, and engineering, including radiative transfer and rarefied gas dynamics.
Abstract: This monograph is intended to be a reasonably self -contained and fairly complete exposition of rigorous results in abstract kinetic theory. Throughout, abstract kinetic equations refer to (an abstract formulation of) equations which describe transport of particles, momentum, energy, or, indeed, any transportable physical quantity. These include the equations of traditional (neutron) transport theory, radiative transfer, and rarefied gas dynamics, as well as a plethora of additional applications in various areas of physics, chemistry, biology and engineering. The mathematical problems addressed within the monograph deal with existence and uniqueness of solutions of initial-boundary value problems, as well as questions of positivity, continuity, growth, stability, explicit representation of solutions, and equivalence of various formulations of the transport equations under consideration. The reader is assumed to have a certain familiarity with elementary aspects of functional analysis, especially basic semigroup theory, and an effort is made to outline any more specialized topics as they are introduced. Over the past several years there has been substantial progress in developing an abstract mathematical framework for treating linear transport problems. The benefits of such an abstract theory are twofold: (i) a mathematically rigorous basis has been established for a variety of problems which were traditionally treated by somewhat heuristic distribution theory methods; and (ii) the results obtained are applicable to a great variety of disparate kinetic processes. Thus, numerous different systems of integrodifferential equations which model a variety of kinetic processes are themselves modelled by an abstract operator equation on a Hilbert (or Banach) space.
TL;DR: In this article, the steady behavior of a gas in contact with its condensed phase of arbitrary shape is investigated on the basis of kinetic theory, and two simple examples (evaporation from a sphere, two-surface problem of evaporation and condensation) are worked out.
Abstract: The steady behavior of a gas in contact with its condensed phase of arbitrary shape is investigated on the basis of kinetic theory. The Knudsen number of the system (the mean free path of the gas molecules divided by the characteristic length of the system) being assumed to be fairly small, the hydrodynamic equations for the macroscopic quantities, the velocity, temperature, and pressure, of the gas and their boundary conditions on the interface of the gas and its condensed phase are derived, and two simple examples (evaporation from a sphere, two-surface problem of evaporation and condensation) are worked out.
TL;DR: In this article, the half space boundary value problem for the Boltzmann equation with an incoming distribution was studied and the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero.
Abstract: In the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero. We linearize it about a drifting Maxwellian and prove that, as conjectured by Cercignani , the problem is well-posed when the drift velocity u exceeds the sound speed c, but that one (respectively four, five) additional conditions must be imposed when 0 < u < c (respectively - c < u < 0 and u < - c). In the second part, we show that the well-posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions.