Download or read book Deterministic Numerical Methods for the Boltzmann Equation written by Simon Pintarelli and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Numerics for the Boltzmann Equation written by Sergej Rjasanow and published by Springer Science & Business Media. This book was released on 2005-11-04 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Deterministic Solvers for the Boltzmann Transport Equation written by Sung-Min Hong and published by Springer Science & Business Media. This book was released on 2011-07-31 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Deterministic Numerical Methods for Unstructured Mesh Neutron Transport Calculation written by Liangzhi Cao and published by Woodhead Publishing. This book was released on 2020-08-30 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows written by V.V. Aristov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Deterministic Numerical Simulation of the Boltzmann and Kinetic Model Equations for Classical and Quantum Dilute Gases written by Lei Wu and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Landau Equation Boltzmann Type Equations Discrete Models and Numerical Methods written by Alexander V Bobylev and published by . This book was released on 2024-09-23 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Deterministic Solvers for the Boltzmann Transport Equation written by Sung-Min Hong and published by Springer. This book was released on 2011-07-31 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Accelerating Solution of the Boltzmann Equation Using Neural Networks written by Thomas Nguyen (Graduate student) and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Lecture Notes on the Discretization of the Boltzmann Equation written by N. Bellomo and published by World Scientific. This book was released on 2003 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book On Study of Deterministic Conservative Solvers for the Nonlinear Boltzmann and Landau Transport Equations written by Chenglong Zhang and published by . This book was released on 2014 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of "shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a "collision matrix" The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided.

Download or read book Landau Equation Boltzmann type Equations Discrete Models and Numerical Methods written by Alexander V. Bobylev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2024-09-23 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book A Deterministic Particle method Solving the Linearized Boltzmann Equation written by Thomas Geyer and published by . This book was released on 1988 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Numerical Solution to the Angle integrated Boltzmann Equation written by J. H. Bennett and published by . This book was released on 1958 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Lecture Notes On The Discretization Of The Boltzmann Equation written by Nicola Bellomo and published by World Scientific. This book was released on 2003-01-24 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Handbook of Numerical Methods for Hyperbolic Problems written by Remi Abgrall and published by Elsevier. This book was released on 2017-01-16 with total page 612 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. Provides detailed, cutting-edge background explanations of existing algorithms and their analysis Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage

Download or read book Uncertainty Quantification for Hyperbolic and Kinetic Equations written by Shi Jin and published by Springer. This book was released on 2018-03-20 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.