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In rarefied and non-continuum conditions, gas is best described at the molecular level and the most physically accurate model is due to the Boltzmann equation. The complexity of the Boltzmann equation suggests that solutions to applications arising in engineering and physics can only be computed numerically. However, solving the Boltzmann equation is extremely difficult because of the high dimensionality of the equation and the high computational costs of evaluation of the collision integral. The objective of this thesis is to accelerate evaluation of the collision integral by replacing deterministic Gauss quadratures in the nodal-DG discretizations of the collision operator with Korobov quadratures. We developed and implemented an algorithm for computing the Boltzmann collision operator using Korobov integration. Accuracy of the multidimensional quadrature formulas was investigated on tests in idealized settings where the exact solutions are known. The method was implemented in FORTRAN. Results of evaluation of the collision operator using Korobov integration was compared to results of evaluation using full tensor product Gauss quadratures.
Gas flows in hyper-sonic air breathing engines and rocket thrusters and flows of particles into vacuum contain regions where the distribution of particle velocities deviates significantly from the Maxwellian distribution. The gas in these regions is said to be in the non-continuum state and its evolution is best described using kinetic equations. The most physically accurate model of non-continuum gas is given by the Boltzmann kinetic equation. However, because of the very high computational costs associated with the evaluation of the collision integral, solution of the Boltzmann equation is used sparingly. Simpler approximate models are often desired in simulations of non-continuum flows in multidimensional applications. However, validating simpler models may be not easy when experimental data is not available. The goal of thesis is to develop novel capabilities for high-fidelity simulation of non-continuum flows with guaranteed accuracy. We will address the our goal by developing fast solvers for the Boltzmann equation using sparse Galerkin approximations of the velocity distribution function and by evaluating accuracy of these solvers using parallel fully deterministic discontinuous Galerkin (DG) Boltzmann solver.
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.
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.
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.
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. Contents: From the Boltzmann Equation to Discretized Kinetic Models (N Bellomo & R Gatignol); Discrete Velocity Models for Gas Mixtures (C Cercignani); Discrete Velocity Models with Multiple Collisions (R Gatignol); Discretization of the Boltzmann Equation and the Semicontinuous Model (L Preziosi & L Rondoni); Semi-continuous Extended Kinetic Theory (W Koller); Steady Kinetic Boundary Value Problems (H Babovsky et al.); Computational Methods and Fast Algorithms for the Boltzmann Equation (L Pareschi); Discrete Velocity Models and Dynamical Systems (A Bobylev & N Bernhoff); Numerical Method for the Compton Scattering Operator (C Buet & S Cordier); Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space (F Schrrer). Readership: Higher level postgraduates in applied mathematics.
The ongoing trend for high-frequency (HF) applications drives the development of high-speed devices. Therefore, trustworthy device simulations are inevitable for understanding and designing future HF devices. During the last decade, the predictive capabilities of Drift-Diffusion (DD) and Hydrodynamic (HD) transport models turned out to be insufficient for state-of-the-art high-frequency transistors. Consequently, a more physics based transport model helps to counter these issues and thus, the Boltzmann transport equation (BTE) comes into focus. In this thesis, a deterministic solution method for the BTE is pursued. First, physical fundamentals and mathematical preconsiderations for the treatment of the BTE are reviewed. This covers the calculation of band structures/dispersion relations, an overview of scattering mechanisms and a detailed description of the coordinate transformations required for analyzing prominent semiconducting materials, such as Silicon-Germanium and III-V compounds, like Indium-Phosphide. The second part focuses on the numerical treatment of the BTE. Besides the employed normalization strategy, the discretization of the BULK BTE is described in detail. Based on the latter, the extensions for the device BTE are specified. A method for the direct calculation of stationary BTE solutions - for the BULK and device case - is introduced and an overview of the WENO method is outlined. The third part is dedicated to the applications of the deterministic solution method and simulation results of the BTE. Recipes for calculating the most important quantities, like current/electron densities, are given. Simulation results for the BULK case and for hetero-junction bipolar transistors are presented and analyzed. Here, the focus is put on both Silicon/Silicon-Germanium and Indium-Phosphide/Indium-Gallium-Arsenide material systems. The part is concluded by a critical review on the current field of application. A summary and an outlook on future extensions
The articles collected in this volume are based on lectures given at the IMA Workshop, "Computational Radiology and Imaging: Therapy and Diagnostics", March 17-21, 1997. Introductory articles by the editors have been added. The focus is on inverse problems involving electromagnetic radiation and particle beams, with applications to X-ray tomography, nuclear medicine, near-infrared imaging, microwave imaging, electron microscopy, and radiation therapy planning. Mathematical and computational tools and models which play important roles in this volume include the X-ray transform and other integral transforms, the linear Boltzmann equation and, for near-infrared imaging, its diffusion approximation, iterative methods for large linear and non-linear least-squares problems, iterative methods for linear feasibility problems, and optimization methods. The volume is intended not only for mathematical scientists and engineers working on these and related problems, but also for non-specialists. It contains much introductory expository material, and a large number of references. Many unsolved computational and mathematical problems of substantial practical importance are pointed out.