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Über dieses Buch

IMA Volumes 135: Transport in Transition Regimes and 136: Dispersive Transport Equations and Multiscale Models focus on the modeling of processes for which transport is one of the most complicated components. This includes processes that involve a wide range of length scales over different spatio-temporal regions of the problem, ranging from the order of mean-free paths to many times this scale. Consequently, effective modeling techniques require different transport models in each region. The first issue is that of finding efficient simulations techniques, since a fully resolved kinetic simulation is often impractical. One therefore develops homogenization, stochastic, or moment based subgrid models. Another issue is to quantify the discrepancy between macroscopic models and the underlying kinetic description, especially when dispersive effects become macroscopic, for example due to quantum effects in semiconductors and superfluids. These two volumes address these questions in relation to a wide variety of application areas, such as semiconductors, plasmas, fluids, chemically reactive gases, etc.

Inhaltsverzeichnis

Frontmatter

BGK-Burnett Equations: A New Set of Second-Order Hydrodynamic Equations for Flows in Continuum-Transition Regime

Abstract
This paper presents a review of the development of a novel set of second-order hydrodynamic equations, designated as the BGK-Burnett equations for computing flows in the continuum-transition regime. The second-order distribution function that forms the basis of this formulation is obtained by the first three terms of the Chapman-Enskog expansion applied to the Boltzmann equation with Bhatnagar-Gross-Krook (BGK) approximation to the collision terms. Such a distribution function, however, does not readily satisfy the moment closure property. Hence, an exact closed form expression for the distribution function is obtained by enforcing moment closure and solving a system of algebraic equations to determine the closure coefficients. Through a series of conjectures, the closure coefficients are designed to move the resulting system of hydrodynamic equations towards an entropy consistent set. An important step in the formulation of the higher-order distribution functions is the proper representation of the material derivatives in terms of the spatial derivatives. While the material derivatives in the first-order distribution function are approximated by Euler Equations, proper representations of these derivatives in the second-order distribution function are determined by an entropy consistent relaxation technique. The BGK-Burnett equations, obtained by taking moments of the Boltzmann equation with the second-order distribution function, are shown to be stable to small wavelength disturbances and entropy consistent for a wide range of grid points and Mach numbers. The paper also describes other equations of Burnett family namely the original, conventional and augmented Burnett equations for the purpose of comparison with BGK-Burnett equations and discusses their shortcomings. The relationship between the Burnett equations and the Grad’s 13 moment equations as shown by Struchtrup by employing the Maxwell-Truesdell-Green iteration is also presented.
Ramesh K. Agarwal

Steady States for Streater’s Energy-Transport Models of Self-Gravitating Particles

Abstract
We review Streater’s energy-transport models which describe the temporal evolution of the density and temperature of a cloud of gravitating particles, coupled to a mean field Poisson equation. In particular we consider the existence of stationary solutions in a bounded domain with given energy and mass. We discuss the influence of the dimension and geometry of the domain on existence results.
Piotr Biler, Jean Dolbeault, Maria J. Esteban, Peter A. Markowich, Tadeusz Nadzieja

Towards a Hybrid Monte Carlo Method for Rarefied Gas Dynamics

Abstract
For the Boltzmann equation, we present a hybrid Monte Carlo method that is robust in the fluid dynamic limit. The method is based on representing the solution as a convex combination of a non-equilibrium particle distribution and a Maxwellian. The hybrid distribution is then evolved by Monte Carlo with an unconditionally stable and asymptotic preserving time discretization. Some computational simulations of spatially homogeneous problems are presented here and extensions to spatially non homogeneous situations discussed.
Russel E. Caflisch, Lorenzo Pareschi

Comparison of Monte Carlo and Deterministic Simulations of a Silicon Diode

Abstract
Electron transport models in Si transistors with channel length of 0.4 microns and 50 nanometers are examined and compared between classical Direct Monte Carlo Simulations and deterministic WENO solvers for a self-consistent kinetic field-relaxation Poisson model. This model is a well accepted low density reduction of the full non-equilibrium transport phenomena. In this comparison we control the calibration of the field dependent, saturated mobility. Our computations show that, at channel length of order 0.4 microns, the relaxation model captures the the first two moments of the particle distribution function inside the channel. In particular a domain decomposition technique that implements classical drift diffusion in the high density regions and augmented drift diffusion inside the channel region gives a correction to the classical drift diffusion simulations, and produces similar qualitative results to the Monte Carlo simulations with a 0.002 CPU time reduction factor. However, we show that in the case of a 50 nanometer channel, the kinetic field-relaxation model fails to approximate well even the first moment, and in particular it does not approximate weil the current voltage curve output from Monte Carlo simulations, making it necessary to incorporate high energy effects into the collision operator.
Jose A. Carrillo, Irene M. Gamba, Orazio Muscato, Chi-Wang Shu

Discrete-Velocity Models for Numerical Simulations in Transitional Regime for Rarefied Flows and Radiative Transfer

Abstract
In this paper we propose a deterministic eulerian approach for kinetic relaxation models. This approach is based on adefinition of the discrete equilibrium using a discrete equivalent of the minimum entropy principle. This leads to discrete models which are entropic and conservative. Two applications to gas dynamics and to radiative transfer are presented. Numerical experiments illustrate the performance of numerical codes based on this approach.
Pierre Charrier, Bruno Dubroca, Luc Mieussens, Rodolphe Turpault

Some Recent Results on the Kinetic Theory of Phase Transitions

Abstract
This paper gives a summary of our recent work on the Becker-Döring (BD) and Lifshitz-Slyozov (LS) models. The main issues addressed are existence, uniqueness and large time behavior for the LS system, and its connection with the BD system. More precisely, we show how the BD system may be rescaled so as to yield the LS system as a first order approximation. Finally by carrying the expansion to second order, we propose a diffusive correction to the classical LS system, which has the form of a Fokker-Planck equation.
Jean-François Collet, Thierry Goudon, Sara Hariz, Frederic Poupaud, Alexis Vasseur

Fluids with Multivalued Internal Energy: The Anisotropic Case

Abstract
This work is concerned with some extensions of the classical compressible Euler model of fluid dynamics in which the fluid internal energy is a measure-valued quantity. A first extension has been derived from the hydrodynamil limit of a kinetic model involving a specific class of collision operators [1,3]. In these papers the collision operator simply describes the isotropization of the kinetic distribution function about some averaging velocity. In the present work we present a new extension of such models in which the relaxed distribution is anisotropic. Similarly to [1] and [3] this model is derived from a kinetic equation with a collision operator that relaxes to anisotropic equilibria. We then investigate diffusive corrections of this fluid dynamical model using Chapman-Enskog techniques and show how the anisotropic character affects the expression of the viscosity and of the heat flux. We argue why such a feature could be used as a tool towards an understanding of fluid turbulence from kinetic theory.
P. Degond, M. Lemou, J. L. Lòpez

A Note on the Energy-Transport Limit of the Semiconductor Boltzmann Equation

Abstract
In this paper, we present a new scaling limit of the semiconductor Boltzmann equation which yields the so-called Energy-Transport model. This model consists of a set of continuity equations for the density and energy together with constitutive relations for the particle and energy fluxes. These fluxes are expressed in terms of gradients of the entropic variables, through a diffusivity matrix related to the Boltzmann collision operators. The present work is devoted to a scaling limit in which the only operator involved in the definition of the diffusivity is the elastic collision operator. Previous derivations required a two-step procedure, resorting to an intermediate model, the so-called Spherical Harmonics Expansion (or SHE) model. We shall present and review the relationship between all these models.
Pierre Degond, C. David Levermore, Christian Schmeiser

Generalized Hydrodynamics and Irreversible Thermodynamics

Abstract
The classical hydrodynamics described by the Navier-Stokes and Fourier equations works impressively if the Mach and Knudsen numbers are low. However, as the gas density becomes sufficiently low as to make the Knudsen number high or as the Mach number becomes large, the classical hydrodynamic theory based on the Navier-Stokes and Fourier equations is found generally inadequate, and it is necessary to improve the theory. The improvement is often sought with the help of the kinetic theory of gases by using the Boltzmann equation or its suitable modifications. The higher order Chapman-Enskog solutions [1] and the Grad moment method [2] have served as the theoretical tools frequently used for the purpose of modifying the classical hydrodynamic theory. There are some mathematically motivated methods [3] developed for the aforementioned goal in relatively recent years.
Byung Chan Eu

A Steady-State Capturing Method for Hyperbolic Systems with Geometrical Source Terms

Abstract
We propose a simple numerical method for capturing the steady state solution of hyperbolic systems with geometrical source terms. We use the interface value, rather than the cell-averages, for the source terms that balance the nonlinear convection at the cell interface, allowing the numerical capturing of the steady state with a formal high order accuracy. This method applies to Godunov or Roe type upwind methods but requires no modification of the Riemann solver. Numerical experiments on scalar conservation laws and the one dimensional shallow water equations show much better resolution of the steady state than the conventional method, with almost no new numerical complexity.
Shi Jin

Maximum Entropy Moment Problems and Extended Euler Equations

Abstract
The reduction of kinetic equations to moment systems leads to a closure problem because material laws have to be expressed in terms of the moment variables. In the maximum entropy approach, the closure problem is solved by assuming that the kinetic distribution function maximizes the entropy under some constraints. For the case of Boltzmann equation, the resulting hyperbolic moment systems are investigated. It turns out that the systems generally have non-convex domains of definition. Moreover, the equilibrium state is typically located on the boundary of the domain of definition where the fluxes are singular. This leads to the strange property that arbitrarily close to equilibrium the characteristic velocities of the moment system can be arbitrarily large.
Michael Junk

Numerical Methods for Radiative Heat Transfer in Diffusive Regimes and Applications to Glass Manufacturing

Abstract
In this paper, different approaches for the numerical solution of radiative heat transfer problems in diffusive regimes are considered. We discuss asymptotic preserving schemes, domain decomposition methods and the development of improved diffusion approximations. Problems related to glass manufacturing processes are numerically investigated.
Axel Klar, Guido Thömmes

Hydrodynamic Limits of the Boltzmann Equation

Abstract
From a physical point of view, we expect that a gas can be described by a fluid mechanic equation when the mean free path goes to zero. We present here some (rigorous) derivation of incompressible Fluid Mechanic equations starting from the Boltzmann equation in the limit where the free mean path (Knudsen number) goes to zero. This work can be seen as an extension of the important series of papers by C. Bardos, F. Golse and D. Levermore [1].
Nader Masmoudi

Sobolev Norm and Carrier Transport in Semiconductors

Abstract
Sobolev norm in H -2 is used to measure the distance between Maxwellian-like distribution functions and particle sets generated by Direct Simulation Monte Carlo method in bulk silicon. This norm can be used as a criteria to determine the domain of validity of the kinetic/macroscopic approaches.
Orazio Muscato

The Evolution of a Gas in a Radiation Field from a Kinetic Point of View

Abstract
An existence theorem is derived for a system of kinetic equations describing the evolution of a gas in a radiation field from a kinetic point of view. The geometrical setting is the slab and given indata. The photons ingoing distribution functions are Dirac measures.
A. Nouri

Hybrid Particle-Based Approach for the Simulation Of Semiconductor Devices: The Full-Band Cellular Automaton/Monte Carlo Method

Abstract
We present a review of the particle-based methods used for the simulation of semiconductor devices. A full-band hybrid cellular automaton/Monte Carlo code far simulation of electron and hole transport in Si and GaAs is presented as well. In this implementation, the entire first Brillouin zone is discretized using a non-uniform mesh in k-space, and a transition table is generated between all initial and final states on some regions of the Brillouin zone, greatly simplifying the final state selection of the conventional Monte Carlo algorithm. Together with its impressive performance, this method allows for fully anisotropic scattering rates within the full-band scheme, at the cost of increased memory requirements for the transition table itself. Good agreement is obtained between the hybrid model and previously reported results for the velocity-field characteristics and high field distribution function, which illustrate the potential accuracy of the technique. Adaptation of the algorithm for parallel computing is discussed as weil.
Marco Saraniti, Shela J. Wigger, Stephen M. Goodnick

Some Remarks on the Equations of Burnett and Grad

Abstract
Both, Grad and Burnett, derived sets of equations from the Boltzmann equation, which improve the classicallaws of Navier-Stokes and Fourier for the description of rarefied gases, i.e. gases with Knudsen numbers above 0.01. Using results of other authors, it is shown that both sets of equations are closer related then is commonly thought - indeed, the Burnett equations can be derived from Grad’s equations by the so-called Maxwellian iteration. This derivation allows to identify the proper form of the Burnett equations in non-inertial frames. Moreover, Grad’s equations with more than 13 moments can describe linear boundary layers while these are not among the phenomena which can be described by Burnett’s equations.
Henning Struchtrup

Boundary Conditions and Boundary Layers for a Class of Linear Relaxation Systems in a Quarter Plane

Abstract
We consider the initial-boundary value problem for the linear one-dimensional Jin-Xin relaxation model in a quarter plane. Our main interest is to understand the boundary layer behavior of the solution and its asymptotic convergence to the corresponding equilibrium system of hyperbolic conservation laws. We identify and rigorously justify a necessary and sufficient condition (which we refer to as Stiff Kreiss Condition) on the boundary condition to guarantee the uniform well-posedness of the initial-boundary value problem for the relaxation system independent of the relaxation parameter. The Stiff Kreiss Condition is derived by using a normal mode analysis. The asymptotic convergence and boundary layer behavior are studied by the Laplace transform and a matched asymptotic analysis. An optimal rate of convergence is obtained.
Zhouping Xin, Wen-Qing Xu

Backmatter

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