Download or read book Reduced Basis Method for Boltzmann Equation written by Revanth Reddy Garlapati and published by . This book was released on 2006 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main aim of the project is to solve the BGK model of the Knudsen parameterized Boltzmann equation which is 1-d with respect to both space and velocity. In order to solve the Boltzmann equation, we first transform the original differential equation by replacing the dependent variable with another variable, weighted with function t(y); next we obtain a Petrov Galerkin weak form of this new transformed equation. To obtain a stable and accurate solution of this weak form, we perform a transformation of the velocity variable y, such that the semi-infinite domain is mapped into a finite domain; we choose the weighting function t(y), to balance contributions at infinity. Once we obtain an accurate and well defined finite element solution of the problem. The next step is to perform the reduced basis analysis of the equation using these accurate finite element solutions. We conclude the project by verifying that the orthonormal reduced Basis method based on the greedy algorithm converges rapidly over the chosen test space.
Download or read book Fast Solution of the Poisson Boltzmann Equation by the Reduced Basis Method and Range separated Canonical Tensor Format written by Cleophas Muganda Kweyu and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Certified Reduced Basis Methods for Parametrized Partial Differential Equations written by Jan S Hesthaven and published by Springer. This book was released on 2015-08-20 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.
Download or read book Separated Representations and PGD Based Model Reduction written by Francisco Chinesta and published by Springer. This book was released on 2014-09-02 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers in this volume start with a description of the construction of reduced models through a review of Proper Orthogonal Decomposition (POD) and reduced basis models, including their mathematical foundations and some challenging applications, then followed by a description of a new generation of simulation strategies based on the use of separated representations (space-parameters, space-time, space-time-parameters, space-space,...), which have led to what is known as Proper Generalized Decomposition (PGD) techniques. The models can be enriched by treating parameters as additional coordinates, leading to fast and inexpensive online calculations based on richer offline parametric solutions. Separated representations are analyzed in detail in the course, from their mathematical foundations to their most spectacular applications. It is also shown how such an approximation could evolve into a new paradigm in computational science, enabling one to circumvent various computational issues in a vast array of applications in engineering science.
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 Lecture Notes on the Discretization of the Boltzmann Equation written by Nicola Bellomo and published by World Scientific. This book was released on 2003 with total page 320 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. 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.
Download or read book The Boltzmann Equation written by E.G.D. Cohen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 647 pages. Available in PDF, EPUB and Kindle. Book excerpt: In,1872, Boltzmann published a paper which for the first time provided a precise mathematical basis for a discussion of the approach to equilibrium. The paper dealt with the approach to equilibrium of a dilute gas and was based on an equation - the Boltzmann equation, as we call it now - for the velocity distribution function of such ~ gas. The Boltzmann equation still forms the basis of the kinetic theory of gases and has proved fruitful not only for the classical gases Boltzmann had in mind, but als- if properly generalized - for the electron gas in a solid and the excitation gas in a superfluid. Therefore it was felt by many of us that the Boltzmann equation was of sufficient interest, even today, to warrant a meeting, in which a review of its present status would be undertaken. Since Boltzmann had spent a good part of his life in Vienna, this city seemed to be a natural setting for such a meeting. The first day was devoted to historical lectures, since it was generally felt that apart from their general interest, they would furnish a good introduction to the subsequent scientific sessions. We are very much indebted to Dr. D.
Download or read book Lecture Notes on the Mathematical Theory of the Boltzmann Equation written by N. Bellomo and published by World Scientific. This book was released on 1995 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of four lectures on some mathematical aspects related to the nonlinear Boltzmann equation. The following topics are dealt with: derivation of kinetic equations, qualitative analysis of the initial value problem, singular perturbation analysis towards the hydrodynamic limit and computational methods towards the solution of problems in fluid dynamics.
Download or read book Reduced Basis Method for Maxwell s Equations with Resonance Phenomena written by Martin Hammerschmidt and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Tensor Numerical Methods in Scientific Computing written by Boris N. Khoromskij and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-06-11 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: The most difficult computational problems nowadays are those of higher dimensions. This research monograph offers an introduction to tensor numerical methods designed for the solution of the multidimensional problems in scientific computing. These methods are based on the rank-structured approximation of multivariate functions and operators by using the appropriate tensor formats. The old and new rank-structured tensor formats are investigated. We discuss in detail the novel quantized tensor approximation method (QTT) which provides function-operator calculus in higher dimensions in logarithmic complexity rendering super-fast convolution, FFT and wavelet transforms. This book suggests the constructive recipes and computational schemes for a number of real life problems described by the multidimensional partial differential equations. We present the theory and algorithms for the sinc-based separable approximation of the analytic radial basis functions including Green’s and Helmholtz kernels. The efficient tensor-based techniques for computational problems in electronic structure calculations and for the grid-based evaluation of long-range interaction potentials in multi-particle systems are considered. We also discuss the QTT numerical approach in many-particle dynamics, tensor techniques for stochastic/parametric PDEs as well as for the solution and homogenization of the elliptic equations with highly-oscillating coefficients. Contents Theory on separable approximation of multivariate functions Multilinear algebra and nonlinear tensor approximation Superfast computations via quantized tensor approximation Tensor approach to multidimensional integrodifferential equations
Download or read book Reduced basis Methods Applied to Locally Non affine and Locally Non linear Partial Differential Equations written by Yuri Olegovich Solodukhov and published by . This book was released on 2005 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: (Cont.) Numerical results are provided with respect to the accuracy and computational savings provided by the described reduced basis methods.
Download or read book Tensor Numerical Methods in Quantum Chemistry written by Venera Khoromskaia and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-06-11 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 1015. Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.
Download or read book The Boltzmann Equation and Its Applications written by Carlo Cercignani and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical mechanics may be naturally divided into two branches, one dealing with equilibrium systems, the other with nonequilibrium systems. The equilibrium properties of macroscopic systems are defined in principle by suitable averages in well-defined Gibbs's ensembles. This provides a frame work for both qualitative understanding and quantitative approximations to equilibrium behaviour. Nonequilibrium phenomena are much less understood at the present time. A notable exception is offered by the case of dilute gases. Here a basic equation was established by Ludwig Boltzmann in 1872. The Boltzmann equation still forms the basis for the kinetic theory of gases and has proved fruitful not only for a study of the classical gases Boltzmann had in mind but also, properly generalized, for studying electron transport in solids and plasmas, neutron transport in nuclear reactors, phonon transport in superfluids, and radiative transfer in planetary and stellar atmospheres. Research in both the new fields and the old one has undergone a considerable advance in the last thirty years.
Download or read book Fast Evaluation of Time harmonic Maxwell s Equations Using the Reduced Basis Method written by Martin W. Hess and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: The reduced basis method (RBM) generates low order models for the solution of parametrized partial differential equations (PDEs) to allow for efficient evaluation in many-query and real-time contexts. We show the theoretical framework in which the RBM is applied to Maxwell\'s equations and present numerical results for model reduction in frequency domain. Using rigorous error estimators, the RBM achieves low order models under variation of material parameters and geometry. The RBM reduces model order by a factor of 50 to 100 and reduces compute time by a factor of 200 and more for numerical experiments using standard circuit elements.
Download or read book Parameter Space The Final Frontier Certified Reduced Basis Methods for Real Time Reliable Solution of Parametrized Partial Differential Equations written by and published by . This book was released on 2007 with total page 13 pages. Available in PDF, EPUB and Kindle. Book excerpt: This project is focused on reduced basis approximation methods, associated rigorous and sharp aposteriori error bounds, and offline-online computational strategies for the rapid and reliable solution of parametrized elliptic, parabolic, and more recently hyperbolic partial differential equations relevant to mechanics from the quantum through the meso-scale to the macro-scale. Typical equations and applications of interest include Density Functional Theory for solid state property calculations, the Boltzmann equation for microscale gas flows, the Navier-Stokes equations for natural convection calculations, elasticity for stress intensity factors/brittle failure, and Helmholtz and the wave equation for acoustic waveguide applications. Of particular interest is real-time and robust parameter estimation with application to detection, nondestructive evaluation, adaptive design/optimization, and control. In the online/deployed stage, we can provide results for key engineering outputs in real-time without loss of accuracy or reliability: the outputs provided - in milliseconds (online) - by our approach are provably indistinguishable from the outputs provided - typically in many minutes or even hours - by classical methods.
Download or read book Computational Fluid Dynamics written by Jiyuan Tu and published by Elsevier. This book was released on 2023-05-09 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Fluid Dynamics: A Practical Approach, Fourth Edition is an introduction to computational fluid dynamics (CFD) fundamentals and commercial CFD software to solve engineering problems. The book is designed for a wide variety of engineering students new to CFD, but is also ideal for practicing engineers learning CFD for the first time. Combining an appropriate level of mathematical background, worked examples, computer screen shots, and step-by-step processes, this book walks the reader through modeling and computing, as well as interpreting CFD results. This new edition has been updated throughout, with new content and improved figures, examples and problems. - Updated throughout, with new case studies, examples, references, and corrections according to readers' and reviewers' feedback - Delivers the latest developments in CFD including the high-order and reduced-order modeling approach, machine learning–accelerated CFD, full coverage of high-speed fluid dynamics, and the meshless approaches to provide a broader overview of the application areas where CFD can be used - Reorganized and rewritten to better meet the needs of CFD instructors and students - Online resources include all lecturing and guest lecturing PPTs, computer lab practicing with step-by-step and screenshot guidelines, assignment and course project details, answers for review questions in each chapter, a new bonus chapter featuring detailed case studies, and result discussion
Download or read book Certified Reduced Basis Methods for Nonaffine Linear Time varying Partial Differential Equations written by Martin A. Grepl and published by . This book was released on 2011 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:
