Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

http://dsec.pku.edu.cn/~tieli/notes/numer_anal/MCQMC_Caflisch.pdf

Monte Carlo and quasi-Monte Carlo methods

This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions. The closure proceeds in two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit. The second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct Navier-Stokes behavior. This is achieved through the introduction of a generalization of the BGK collision operator. The simplest such system in three spatial dimensions is a “14-moment” closure, which also recovers the behavior of the Grad “13-moment” system when the velocity distributions lie near local Maxwellians. The closure procedure can be applied to a general class of kinetic theories.

Moment Closure Hierarchies for Kinetic Theories

We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N × N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 × 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 × 2 case. © 1994 John Wiley & Sons, Inc.

Hyperbolic conservation laws with stiff relaxation terms and entropy

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A kinetic formulation of multidimensional scalar conservation laws and related equations

The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.

https://www.ljll.math.upmc.fr/~bardos/pdf/kinederiv.pdf

Fluid dynamic limits of kinetic equations. I. Formal derivations

In highly scattering regimes, the transport equation has a limit in which the leading behavior of its solution is determined by the solution of a diffusion equation. A boundary layer with a thickness of a few mean free paths usually forms between the domain boundary and the interior region, through which the given transport boundary conditions are matched to the interior tablesolution by properly choosing the boundary conditions for the diffusion equation. In order for a numerical scheme to be effective in these regimes, it must have both a correct interior diffusion limit and a correct boundary condition limit. The behavior of the discrete-ordinate method is studied in these limits and formulas for the resulting diffusion equation and its boundary conditions are found. By imposing the condition that these limiting formulas be the same as those for the transport equation, a new constraint on the quadrature set is obtained and a way to discretize the boundary data based on the discrete W-function ...

The discrete-ordinate method in diffusive regimes

We present, for a general class of kinetic equations, asymptotic limits leading to various equations of incompressible fluid mechanics: the Navier-Stokes equations, the linearized Navier-Stokes equations, the Euler equations and the linearized Euler equations. For all of these limits, we also obtain the Boussinesq approximation and an equation governing the temperature. For Di Perna-Lions solutions of the Boltzmann equation [1], we state a convergence theorem for the linearized Navier-Stokes limit, as well as a partial result the nonlinear Navier-Stokes case Cette note presente, pour une classe assez generale d'equations cinetiques, des limites asymptotiques conduisant a diverses equations de la Mecanique des fluides incompressibles, parmi lesquelles: l'equation de Navier-Stokes, l'equation de Navier-Stokes linearisee, l'equation d'Euler et l'equation d'Euler linearisee. Pour chacune de ces limites, nous obtenons aussi l'approximation de Boussinesq ainsi qu'une equation regissant la temperature. Nous enoncons pour les solutions «a la Di Perna-Lions» de l'equation de Boltzmann [1], un theoreme de convergence vers l'equation de Navier-Stokes linearisee, ainsi qu'un resultat partiel analogue pour l'equation de Navier-Stokes non lineaire

Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles

The connection between discrete velocity kinetic theory and fluid dynamics is systematically described. Conditions that formally lead to generalized compressible Euler equations or to generalized incompressible Navier-Stokes equations are given. These conditions are related to an H-theorem. It is proven that a large class of polynomial collision operators in semidetailed balance satisfies this H-theorem. Finally, results are given concerning the global validity in time of the convergence for the case where the formal scaling of the kinetic equation leads to the linearized incompressible Navier-Stokes limit.

Fluid Dynamic Limits of Discrete Velocity Kinetic Equations

The connection between kinetic theory and the macroscopic equations of fluid dynamics is described In particular, our results concerning the incompressible Navier-Stokes equation are compared with the classical derivation of Hilbert and Chapman-Enskog Some indications of the validity of these limits are given More specifically, the connection between the DiPerna-Lions renormalized solution for the Boltzmann equation and the Leray-Hopf solution for the Navier-Stokes equation is considered

David Levermore

Papers

Fluid dynamic limits of kinetic equations. I. Formal derivations

The discrete-ordinate method in diffusive regimes

Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles

Fluid Dynamic Limits of Discrete Velocity Kinetic Equations

Macroscopic Limits of Kinetic Equations