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Author : Simon Pintarelli
Publisher :
Page : pages
File Size : 36,65 MB
Release : 2018
Category :
ISBN :
Author : Sergej Rjasanow
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 47,53 MB
Release : 2005-11-04
Category : Mathematics
ISBN : 3540276890
Stochastic numerical methods play an important role in large scale computations in the applied sciences. The first goal of this book is to give a mathematical description of classical direct simulation Monte Carlo (DSMC) procedures for rarefied gases, using the theory of Markov processes as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method. This method includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous convergence results as well as detailed numerical studies are presented.
Author : Sung-Min Hong
Publisher : Springer Science & Business Media
Page : 235 pages
File Size : 21,61 MB
Release : 2011-07-31
Category : Technology & Engineering
ISBN : 3709107784
The book covers all aspects from the expansion of the Boltzmann transport equation with harmonic functions to application to devices, where transport in the bulk and in inversion layers is considered. The important aspects of stabilization and band structure mapping are discussed in detail. This is done not only for the full band structure of the 3D k-space, but also for the warped band structure of the quasi 2D hole gas. Efficient methods for building the Schrödinger equation for arbitrary surface or strain directions, gridding of the 2D k-space and solving it together with the other two equations are presented.
Author : Liangzhi Cao
Publisher : Woodhead Publishing
Page : 294 pages
File Size : 16,77 MB
Release : 2020-08-30
Category : Technology & Engineering
ISBN : 0128182229
Deterministic Numerical Methods for Unstructured-Mesh Neutron Transport Calculation presents the latest deterministic numerical methods for neutron transport equations (NTEs) with complex geometry, which are of great demand in recent years due to the rapid development of advanced nuclear reactor concepts and high-performance computational technologies. This book covers the wellknown methods proposed and used in recent years, not only theoretical modeling but also numerical results. This book provides readers with a very thorough understanding of unstructured neutron transport calculations and enables them to develop their own computational codes. The fundamentals, numerical discretization methods, algorithms, and numerical results are discussed. Researchers and engineers from utilities and research institutes are provided with examples on how to model an advanced nuclear reactor, which they can then apply to their own research projects and lab settings. Combines the theoretical models with numerical methods and results in one complete resource Presents the latest progress on the topic in an easy-to-navigate format
Author : V.V. Aristov
Publisher : Springer Science & Business Media
Page : 305 pages
File Size : 20,19 MB
Release : 2012-12-06
Category : Science
ISBN : 9401008663
This book is concerned with the methods of solving the nonlinear Boltz mann equation and of investigating its possibilities for describing some aerodynamic and physical problems. This monograph is a sequel to the book 'Numerical direct solutions of the kinetic Boltzmann equation' (in Russian) which was written with F. G. Tcheremissine and published by the Computing Center of the Russian Academy of Sciences some years ago. The main purposes of these two books are almost similar, namely, the study of nonequilibrium gas flows on the basis of direct integration of the kinetic equations. Nevertheless, there are some new aspects in the way this topic is treated in the present monograph. In particular, attention is paid to the advantages of the Boltzmann equation as a tool for considering nonequi librium, nonlinear processes. New fields of application of the Boltzmann equation are also described. Solutions of some problems are obtained with higher accuracy. Numerical procedures, such as parallel computing, are in vestigated for the first time. The structure and the contents of the present book have some com mon features with the monograph mentioned above, although there are new issues concerning the mathematical apparatus developed so that the Boltzmann equation can be applied for new physical problems. Because of this some chapters have been rewritten and checked again and some new chapters have been added.
Author : Lei Wu
Publisher :
Page : 0 pages
File Size : 18,1 MB
Release : 2013
Category :
ISBN :
In the areas of low-density aerodynamics, vacuum industry, and micro-electromechanical systems, the Navier-Stokes-Fourier equations fail to describe the gas dynamics when the molecular mean free path is not negligible compared to the characteristic flow length. Instead, the Boltzmann equation is used to account for the non-continuum nature of the rarefied gas. Although many efforts have been made to derive the macroscopic equations from the Boltzmann equation, the numerical simulation of the Boltzmann equation is indispensable in the study of moderately and highly rarefied gas. We aim to develop an accurate and efficient deterministic numerical method to solve the Boltzmann equation. The fast spectral method [1], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical results with the exact Bobylev-Krook-Wu solutions of the space-homogeneous Boltzmann equation for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared and the use of special collision kernels is justified. An iteration scheme, where the numerical errors decay exponentially, is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation. Sever classical benchmarking problems (the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows) are investigated. For normal shock waves, our numerical results are compared with the finite-difference solution of the Boltzmann equation for hard sphere molecules, the experimental data, and the molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For the planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the Direct Simulation Monte Carlo method. Excellent agreements are observed in all test cases. The fast spectral method is then applied to the linearised Boltzmann equation. With appropriate velocity discretization, the classical Poiseuille and thermal creep flows are solved up to Kn 106, where the accuracy in the mass and heat flow rates is comparable to those from the finite-difference method and the efficiency is much better than the low-noise Direct Simulation Monte Carlo method. The fast spectral method is also extended to solve the Boltzmann equation for binary gas mixtures, both in the framework of classical and quantum mechanics. With the accurate numerical solution provided by the fast spectral method, we check the accuracy of kinetic model equations to find out at what flow regime can the complicated Boltzmann collision kernel be replaced by the simple kinetic ones. We also solve the collective oscillation of quantum gas confined in external trap and compare the numerical solutions with the experimental data, indicating the applicability of quantum kinetic model.
Author : Alexander V Bobylev
Publisher :
Page : 0 pages
File Size : 25,34 MB
Release : 2024-09-23
Category : Mathematics
ISBN : 9783110550139
This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The second volume covers discrete velocity models of the Boltzmann equation, results on the Landau equation, and numerical (deterministic and stochastic) methods for the solution of kinetic equations.
Author : Sung-Min Hong
Publisher : Springer
Page : 227 pages
File Size : 50,21 MB
Release : 2011-07-31
Category : Technology & Engineering
ISBN : 9783709107775
The book covers all aspects from the expansion of the Boltzmann transport equation with harmonic functions to application to devices, where transport in the bulk and in inversion layers is considered. The important aspects of stabilization and band structure mapping are discussed in detail. This is done not only for the full band structure of the 3D k-space, but also for the warped band structure of the quasi 2D hole gas. Efficient methods for building the Schrödinger equation for arbitrary surface or strain directions, gridding of the 2D k-space and solving it together with the other two equations are presented.
Author : Thomas Nguyen (Graduate student)
Publisher :
Page : 0 pages
File Size : 44,43 MB
Release : 2022
Category :
ISBN :
Different methods have been developed to solve the Boltzmann equation during the past decades: the direct simulation Monte Carlo method, the lattice Boltzmann method, and the direct deterministic methods for computing the Boltzmann equation. However, computational costs of the existing methods are still prohibitive for simulating complex flows in three dimensions and flows of multi-component gases with real gas effects. Methods of increased efficiency need to be proposed in order to continue advancement in these areas. In this thesis, we explore use of neural networks for solving the Boltzmann equation for a class of problems of spatially homogeneous relaxation of sums of two Maxwellian streams. The data set for training the neural networks is generated by solving the Boltzmann equation using classical methods. We consider applications of deep autoencoder to learn a compressed representation of the solution dataset and to filtering of truncation errors in numerical solutions. The Boltzmann collision operator is approximated using deep convolutional neural networks (CNNs). Accuracy of the trained autoencoders and CNNs was investigated. We use the trained CNNs and Euler method to numerically solve the spatially homogeneous Boltzmann equation. The results are compared to solutions obtained by deterministic solvers. The solutions obtained by CNNs showed good agreement with the results obtained by classical methods while providing at least three orders of magnitude acceleration. The computer memory requirements were found to be comparable to requirements of the classical methods. Small violations of conservation of mass and energy are observed as solution are reaching the steady state. Additionally, the solutions appear to be not stable on an infinite time interval. However, both issues can be corrected using established numerical methods for kinetic equations.
Author : N. Bellomo
Publisher : World Scientific
Page : 317 pages
File Size : 26,33 MB
Release : 2003
Category : Science
ISBN : 9812382259
This book presents contributions on the following topics: discretization methods in the velocity and space, analysis of the conservation properties, asymptotic convergence to the continuous equation when the number of velocities tends to infinity, and application of discrete models. It consists of ten chapters. Each chapter is written by applied mathematicians who have been active in the field, and whose scientific contributions are well recognized by the scientific community.