Uniformly Convergent Approximations For Convection Diffusion Problems
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Uniformly Convergent Approximations For Convection Diffusion Problems
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Author : Özgür BINGÖL
language : en
Publisher: LAP Lambert Academic Publishing
Release Date : 2010-05
Uniformly Convergent Approximations For Convection Diffusion Problems written by Özgür BINGÖL and has been published by LAP Lambert Academic Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-05 with categories.
Mathematical models that involve a combination of convective and diffusive processes are among the most widespread in all of science, engineering and other fields where mathematical modelling is important.Water quality problems, convective heat transfer problems, simulation of the semiconductor devices can be given as an example of these models. Also, the Linearization of the Navier-Stokes equation and drift-diffusion equation of semiconductor device modelling are important instances. The dimensionless parameter that measures the relative strenght of diffusion is generally quite small; so one often meets with situations where thin boundary and interior layers are present and singular perturbation problems arise. In this case, the main difficulty is to obtain a numerical solution which converges -Uniformly to the exact solution of the problem. In this work, the numerical approximations of the convection diffusion problem both on a uniform and non-uniform meshes are investigated. Also, it is shown that these numerical approximations have -Uniform convergency. This book will be useful for ones who research on this subject area.
Uniformly Convergent Approximation On Special Meshes
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Author : Özgür Bingöl
language : en
Publisher:
Release Date : 2007
Uniformly Convergent Approximation On Special Meshes written by Özgür Bingöl and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with categories.
We consider finite difference methods for the approximation of one-dimensional convection-diffusion problem with a small parameter multiplying the diffusion term. An analysis of the centered difference and upwind difference schemes on equidistant meshes shows that these methods are not uniformly convergent in the discrete maximum norm. However, we show that the upwind method over a set of suitably distributed mesh points produce uniformly convergent approximations in the discrete maximum norm. We further investigate the upwind difference method for the approximation of the convection-diffusion problem with a point source. Theoretical findings are supported with the numerical results.
Convection Diffusion Problems An Introduction To Their Analysis And Numerical Solution
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Author : Martin Stynes
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-11-21
Convection Diffusion Problems An Introduction To Their Analysis And Numerical Solution written by Martin Stynes and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-21 with Differential equations categories.
Many physical problems involve diffusive and convective (transport) processes. When diffusion dominates convection, standard numerical methods work satisfactorily. But when convection dominates diffusion, the standard methods become unstable, and special techniques are needed to compute accurate numerical approximations of the unknown solution. This convection-dominated regime is the focus of the book. After discussing at length the nature of solutions to convection-dominated convection-diffusion problems, the authors motivate and design numerical methods that are particularly suited to this class of problems. At first they examine finite-difference methods for two-point boundary value problems, as their analysis requires little theoretical background. Upwinding, artificial diffusion, uniformly convergent methods, and Shishkin meshes are some of the topics presented. Throughout, the authors are concerned with the accuracy of solutions when the diffusion coefficient is close to zero. Later in the book they concentrate on finite element methods for problems posed in one and two dimensions. This lucid yet thorough account of convection-dominated convection-diffusion problems and how to solve them numerically is meant for beginning graduate students, and it includes a large number of exercises. An up-to-date bibliography provides the reader with further reading.
Numerical Methods For Singularly Perturbed Differential Equations
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Author : Hans-Görg Roos
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Numerical Methods For Singularly Perturbed Differential Equations written by Hans-Görg Roos and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
The analysis of singular perturbed differential equations began early in this century, when approximate solutions were constructed from asymptotic ex pansions. (Preliminary attempts appear in the nineteenth century [vD94].) This technique has flourished since the mid-1960s. Its principal ideas and methods are described in several textbooks. Nevertheless, asymptotic ex pansions may be impossible to construct or may fail to simplify the given problem; then numerical approximations are often the only option. The systematic study of numerical methods for singular perturbation problems started somewhat later - in the 1970s. While the research frontier has been steadily pushed back, the exposition of new developments in the analysis of numerical methods has been neglected. Perhaps the only example of a textbook that concentrates on this analysis is [DMS80], which collects various results for ordinary differential equations, but many methods and techniques that are relevant today (especially for partial differential equa tions) were developed after 1980.Thus contemporary researchers must comb the literature to acquaint themselves with earlier work. Our purposes in writing this introductory book are twofold. First, we aim to present a structured account of recent ideas in the numerical analysis of singularly perturbed differential equations. Second, this important area has many open problems and we hope that our book will stimulate further investigations.Our choice of topics is inevitably personal and reflects our own main interests.
Uniform Convergence Rates For Eulerian And Lagrangian Finite Element Approximations Of Convection Dominated Diffusion Problems
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Author : Markus Bause
language : de
Publisher:
Release Date : 1999
Uniform Convergence Rates For Eulerian And Lagrangian Finite Element Approximations Of Convection Dominated Diffusion Problems written by Markus Bause and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with categories.
Uniform Convergence Rates For Eulerian And Lagrangian Finite Element Approximations Of Convection Dominated Diffusion Problems
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Author : Markus Bause
language : en
Publisher:
Release Date : 1999
Uniform Convergence Rates For Eulerian And Lagrangian Finite Element Approximations Of Convection Dominated Diffusion Problems written by Markus Bause and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with categories.
Revival Numerical Solution Of Convection Diffusion Problems 1996
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Author : K.W. Morton
language : en
Publisher: CRC Press
Release Date : 2019-02-25
Revival Numerical Solution Of Convection Diffusion Problems 1996 written by K.W. Morton and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-25 with Mathematics categories.
Accurate modeling of the interaction between convective and diffusive processes is one of the most common challenges in the numerical approximation of partial differential equations. This is partly due to the fact that numerical algorithms, and the techniques used for their analysis, tend to be very different in the two limiting cases of elliptic and hyperbolic equations. Many different ideas and approaches have been proposed in widely differing contexts to resolve the difficulties of exponential fitting, compact differencing, number upwinding, artificial viscosity, streamline diffusion, Petrov-Galerkin and evolution Galerkin being some examples from the main fields of finite difference and finite element methods. The main aim of this volume is to draw together all these ideas and see how they overlap and differ. The reader is provided with a useful and wide ranging source of algorithmic concepts and techniques of analysis. The material presented has been drawn both from theoretically oriented literature on finite differences, finite volume and finite element methods and also from accounts of practical, large-scale computing, particularly in the field of computational fluid dynamics.
Discrete Approximations For Singularly Perturbed Boundary Value Problems With Parabolic Layers
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Author : P. A. Farrell
language : en
Publisher:
Release Date : 1995
Discrete Approximations For Singularly Perturbed Boundary Value Problems With Parabolic Layers written by P. A. Farrell and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Boundary value problems categories.
Abstract: "Singularly perturbed boundary value problems for equations of elliptic and parabolic type are studied. For small values of the perturbation parameter, parabolic boundary layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux derived from it do not converge uniformly with respect to this parameter. In particular, the relative error of the diffusive flux becomes unbounded as the perturbation parameter tends to zero. Using the method of special condensing grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges [epsilon]-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. Also for parabolic equations [epsilon]-uniformly convergent approximations for the normalised fluxes are constructed. Results of numerical experiments are discussed. Summarising, we consider: 1. Problems for Singularly Perturbed (SP) parabolic equation with discontinuous boundary conditions. 2. Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type. 3. Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required."
Analytical And Numerical Methods For Convection Dominated And Singularly Perturbed Problems
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Author : Lubin Vulkov
language : en
Publisher: Nova Publishers
Release Date : 2000
Analytical And Numerical Methods For Convection Dominated And Singularly Perturbed Problems written by Lubin Vulkov and has been published by Nova Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
This volume is the Proceedings of the Workshop on Analytical and Computational Methods for Convection-Dominated and Singularly Perturbed Problems, which took place in Lozenetz, Bulgaria, 27-31 August 1998. The workshop attracted about 50 participants from 12 countries. The volume includes 13 invited lectures and 19 contributed papers presented at the workshop and thus gives an overview of the latest developments in both the theory and applications of advanced numerical methods to problems having boundary and interior layers. There was an emphasis on experiences from the numerical analysis of such problems and on theoretical developments. The aim of the workshop was to provide an opportunity for scientists from the East and the West, who develop robust methods for singularly perturbed and related problems and also who apply these methods to real-life problems, to discuss recent achievements in this area and to exchange ideas with a view of possible research co-operation.
Layer Adapted Meshes For Reaction Convection Diffusion Problems
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Author : Torsten Linß
language : en
Publisher: Springer
Release Date : 2009-11-21
Layer Adapted Meshes For Reaction Convection Diffusion Problems written by Torsten Linß and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-11-21 with Mathematics categories.
This is a book on numerical methods for singular perturbation problems – in part- ular, stationary reaction-convection-diffusion problems exhibiting layer behaviour. More precisely, it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. Numerical methods for singularly perturbed differential equations have been studied since the early 1970s and the research frontier has been constantly - panding since. A comprehensive exposition of the state of the art in the analysis of numerical methods for singular perturbation problems is [141] which was p- lished in 2008. As that monograph covers a big variety of numerical methods, it only contains a rather short introduction to layer-adapted meshes, while the present book is exclusively dedicated to that subject. An early important contribution towards the optimisation of numerical methods by means of special meshes was made by N.S. Bakhvalov [18] in 1969. His paper spawned a lively discussion in the literature with a number of further meshes - ing proposed and applied to various singular perturbation problems. However, in the mid 1980s, this development stalled, but was enlivened again by G.I. Shishkin’s proposal of piecewise-equidistant meshes in the early 1990s [121,150]. Because of their very simple structure, they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on c- peting meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue.