Numerical Methods For Singularly Perturbed Differential Equations

Author : Hans-Görg Roos,Martin Stynes,Lutz Tobiska
Publisher : Springer Science & Business Media
Page : 364 pages
File Size : 53,7 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662032060

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Numerical Methods for Singularly Perturbed Differential Equations by Hans-Görg Roos,Martin Stynes,Lutz Tobiska Pdf

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.

Author : Hans-Görg Roos,Martin Stynes,Lutz Tobiska
Publisher : Springer Science & Business Media
Page : 599 pages
File Size : 50,7 Mb
Release : 2008-09-17
Category : Mathematics
ISBN : 9783540344674

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Robust Numerical Methods for Singularly Perturbed Differential Equations by Hans-Görg Roos,Martin Stynes,Lutz Tobiska Pdf

This new edition incorporates new developments in numerical methods for singularly perturbed differential equations, focusing on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics.

Author : John James Henry Miller,Eugene O'Riordan,G. I. Shishkin
Publisher : World Scientific
Page : 191 pages
File Size : 46,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814390736

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Fitted Numerical Methods for Singular Perturbation Problems by John James Henry Miller,Eugene O'Riordan,G. I. Shishkin Pdf

Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an introduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but the theory of why these numerical methods work is far from simple. This book can be used as an introductory text to the theory underpinning fitted mesh methods.

Author : Hans-Görg Roos,Martin Stynes,Lutz Tobiska
Publisher : Unknown
Page : 372 pages
File Size : 43,9 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662032074

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Numerical Methods for Singularly Perturbed Differential Equations by Hans-Görg Roos,Martin Stynes,Lutz Tobiska Pdf

Author : Hans-G. Roos,Martin Stynes,Lutz Tobiska
Publisher : Springer
Page : 604 pages
File Size : 54,7 Mb
Release : 2009-08-29
Category : Mathematics
ISBN : 3540861971

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Robust Numerical Methods for Singularly Perturbed Differential Equations by Hans-G. Roos,Martin Stynes,Lutz Tobiska Pdf

This new edition incorporates new developments in numerical methods for singularly perturbed differential equations, focusing on linear convection-diffusion equations and on nonlinear flow problems that appear in computational fluid dynamics.

Author : Emmanuel Frénod
Publisher : Springer
Page : 126 pages
File Size : 54,7 Mb
Release : 2017-10-05
Category : Mathematics
ISBN : 9783319646688

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Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations by Emmanuel Frénod Pdf

This book presents the classical results of the two-scale convergence theory and explains – using several figures – why it works. It then shows how to use this theory to homogenize ordinary differential equations with oscillating coefficients as well as oscillatory singularly perturbed ordinary differential equations. In addition, it explores the homogenization of hyperbolic partial differential equations with oscillating coefficients and linear oscillatory singularly perturbed hyperbolic partial differential equations. Further, it introduces readers to the two-scale numerical methods that can be built from the previous approaches to solve oscillatory singularly perturbed transport equations (ODE and hyperbolic PDE) and demonstrates how they can be used efficiently. This book appeals to master’s and PhD students interested in homogenization and numerics, as well as to the Iter community.

Author : P. W. Hemker,John James Henry Miller
Publisher : Unknown
Page : 520 pages
File Size : 49,7 Mb
Release : 1979
Category : Mathematics
ISBN : UOM:39015009241061

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Numerical Analysis of Singular Perturbation Problems by P. W. Hemker,John James Henry Miller Pdf

14 lectures by the invited speakers and 14 shorter contributions from the other speakers (pref.)

Author : Lubin Vulkov,John James Henry Miller
Publisher : Nova Publishers
Page : 298 pages
File Size : 49,7 Mb
Release : 2000
Category : Mathematics
ISBN : 1560728485

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Analytical and Numerical Methods for Convection-dominated and Singularly Perturbed Problems by Lubin Vulkov,John James Henry Miller Pdf

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.

Author : Robert E., Jr. O'Malley
Publisher : Springer Science & Business Media
Page : 234 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209775

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Singular Perturbation Methods for Ordinary Differential Equations by Robert E., Jr. O'Malley Pdf

This book results from various lectures given in recent years. Early drafts were used for several single semester courses on singular perturbation meth ods given at Rensselaer, and a more complete version was used for a one year course at the Technische Universitat Wien. Some portions have been used for short lecture series at Universidad Central de Venezuela, West Vir ginia University, the University of Southern California, the University of California at Davis, East China Normal University, the University of Texas at Arlington, Universita di Padova, and the University of New Hampshire, among other places. As a result, I've obtained lots of valuable feedback from students and listeners, for which I am grateful. This writing continues a pattern. Earlier lectures at Bell Laboratories, at the University of Edin burgh and New York University, and at the Australian National University led to my earlier works (1968, 1974, and 1978). All seem to have been useful for the study of singular perturbations, and I hope the same will be true of this monograph. I've personally learned much from reading and analyzing the works of others, so I would especially encourage readers to treat this book as an introduction to a diverse and exciting literature. The topic coverage selected is personal and reflects my current opin ions. An attempt has been made to encourage a consistent method of ap proaching problems, largely through correcting outer limits in regions of rapid change. Formal proofs of correctness are not emphasized.

Author : Adelaida B. Vasil'eva,Valentin F. Butuzov,Leonid V. Kalachev
Publisher : SIAM
Page : 231 pages
File Size : 45,8 Mb
Release : 1995-01-01
Category : Mathematics
ISBN : 9780898713336

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The Boundary Function Method for Singular Perturbed Problems by Adelaida B. Vasil'eva,Valentin F. Butuzov,Leonid V. Kalachev Pdf

This book is devoted solely to the boundary function method, which is one of the asymptotic methods.

Author : Grigory I. Shishkin,Lidia P. Shishkina
Publisher : CRC Press
Page : 409 pages
File Size : 41,8 Mb
Release : 2008-09-22
Category : Mathematics
ISBN : 9780203492413

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Difference Methods for Singular Perturbation Problems by Grigory I. Shishkin,Lidia P. Shishkina Pdf

Difference Methods for Singular Perturbation Problems focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the ε-uniform convergence of these schemes and surveys the latest approaches important for further progress in numerical methods. The first part of the book e

Author : Luminita Barbu,Gheorghe Morosanu
Publisher : Springer Science & Business Media
Page : 231 pages
File Size : 45,6 Mb
Release : 2007-12-14
Category : Mathematics
ISBN : 9783764383312

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Singularly Perturbed Boundary-Value Problems by Luminita Barbu,Gheorghe Morosanu Pdf

This book offers a detailed asymptotic analysis of some important classes of singularly perturbed boundary value problems which are mathematical models for phenomena in biology, chemistry, and engineering. The authors are particularly interested in nonlinear problems, which have gone little-examined so far in literature dedicated to singular perturbations. The treatment presented here combines successful results from functional analysis, singular perturbation theory, partial differential equations, and evolution equations.

Author : Torsten Linß
Publisher : Springer
Page : 331 pages
File Size : 41,7 Mb
Release : 2009-11-21
Category : Mathematics
ISBN : 9783642051340

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Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems by Torsten Linß Pdf

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.

Author : Jens M. Melenk
Publisher : Springer
Page : 326 pages
File Size : 50,7 Mb
Release : 2004-10-20
Category : Mathematics
ISBN : 9783540457817

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hp-Finite Element Methods for Singular Perturbations by Jens M. Melenk Pdf

Many partial differential equations arising in practice are parameter-dependent problems that are of singularly perturbed type. Prominent examples include plate and shell models for small thickness in solid mechanics, convection-diffusion problems in fluid mechanics, and equations arising in semi-conductor device modelling. Common features of these problems are layers and, in the case of non-smooth geometries, corner singularities. Mesh design principles for the efficient approximation of both features by the hp-version of the finite element method (hp-FEM) are proposed in this volume. For a class of singularly perturbed problems on polygonal domains, robust exponential convergence of the hp-FEM based on these mesh design principles is established rigorously.

Author : Vladimir D. Liseikin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 300 pages
File Size : 45,9 Mb
Release : 2018-11-05
Category : Mathematics
ISBN : 9783110941944

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Layer Resolving Grids and Transformations for Singular Perturbation Problems by Vladimir D. Liseikin Pdf

The approach of layer-damping coordinate transformations to treat singularly perturbed equations is a relatively new, and fast growing area in the field of applied mathematics. This monograph aims to present a clear, concise, and easily understandable description of the qualitative properties of solutions to singularly perturbed problems as well as of the essential elements, methods and codes of the technology adjusted to numerical solutions of equations with singularities by applying layer-damping coordinate transformations and corresponding layer-resolving grids. The first part of the book deals with an analytical study of estimates of the solutions and their derivatives in layers of singularities as well as suitable techniques for obtaining results. In the second part, a technique for building the coordinate transformations eliminating boundary and interior layers, is presented. Numerical algorithms based on the technique which is developed for generating layer-damping coordinate transformations and their corresponding layer-resolving meshes are presented in the final part of this volume. This book will be of value and interest to researchers in computational and applied mathematics.