Difference Schemes

Author : Boško S. Jovanović,Endre Süli
Publisher : Springer Science & Business Media
Page : 416 pages
File Size : 43,6 Mb
Release : 2013-10-22
Category : Mathematics
ISBN : 9781447154600

Get Book

Analysis of Finite Difference Schemes by Boško S. Jovanović,Endre Süli Pdf

This book develops a systematic and rigorous mathematical theory of finite difference methods for linear elliptic, parabolic and hyperbolic partial differential equations with nonsmooth solutions. Finite difference methods are a classical class of techniques for the numerical approximation of partial differential equations. Traditionally, their convergence analysis presupposes the smoothness of the coefficients, source terms, initial and boundary data, and of the associated solution to the differential equation. This then enables the application of elementary analytical tools to explore their stability and accuracy. The assumptions on the smoothness of the data and of the associated analytical solution are however frequently unrealistic. There is a wealth of boundary – and initial – value problems, arising from various applications in physics and engineering, where the data and the corresponding solution exhibit lack of regularity. In such instances classical techniques for the error analysis of finite difference schemes break down. The objective of this book is to develop the mathematical theory of finite difference schemes for linear partial differential equations with nonsmooth solutions. Analysis of Finite Difference Schemes is aimed at researchers and graduate students interested in the mathematical theory of numerical methods for the approximate solution of partial differential equations.

Author : John C. Strikwerda
Publisher : Springer
Page : 410 pages
File Size : 42,7 Mb
Release : 1989-09-28
Category : Juvenile Nonfiction
ISBN : UOM:39015059070451

Get Book

Finite Difference Schemes and Partial Differential Equations by John C. Strikwerda Pdf

Author : Allaberen Ashyralyev,Pavel E. Sobolevskii
Publisher : Birkhäuser
Page : 453 pages
File Size : 48,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879224

Get Book

New Difference Schemes for Partial Differential Equations by Allaberen Ashyralyev,Pavel E. Sobolevskii Pdf

This book explores new difference schemes for approximating the solutions of regular and singular perturbation boundary-value problems for PDEs. The construction is based on the exact difference scheme and Taylor's decomposition on the two or three points, which permits investigation of differential equations with variable coefficients and regular and singular perturbation boundary value problems.

Author : Alexander A. Samarskii
Publisher : CRC Press
Page : 796 pages
File Size : 50,8 Mb
Release : 2001-03-29
Category : Mathematics
ISBN : 0203908511

Get Book

The Theory of Difference Schemes by Alexander A. Samarskii Pdf

The Theory of Difference Schemes emphasizes solutions to boundary value problems through multiple difference schemes. It addresses the construction of approximate numerical methods and computer algorithms for solving mathematical physics problems. The book also develops mathematical models for obtaining desired solutions in minimal time using direct or iterative difference equations. Mathematical Reviews said it is "well-written [and] an excellent book, with a wealth of mathematical material and techniques."

Author : S.K. Godunov,V.S. Ryabenkii
Publisher : Elsevier
Page : 488 pages
File Size : 42,9 Mb
Release : 1987-05-01
Category : Mathematics
ISBN : 0080875408

Get Book

Difference Schemes by S.K. Godunov,V.S. Ryabenkii Pdf

Much applied and theoretical research in natural sciences leads to boundary-value problems stated in terms of differential equations. When solving these problems with computers, the differential problems are replaced approximately by difference schemes. This book is an introduction to the theory of difference schemes, and was written as a textbook for university mathematics and physics departments and for technical universities. Some sections of the book will be of interest to computations specialists. While stressing a mathematically rigorous treatment of model problems, the book also demonstrates the relation between theory and computer experiments, using difference schemes created for practical computations.

Author : Petr N. Vabishchevich
Publisher : Walter de Gruyter
Page : 370 pages
File Size : 49,8 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 9783110321463

Get Book

Additive Operator-Difference Schemes by Petr N. Vabishchevich Pdf

Applied mathematical modeling is concerned with solving unsteady problems. Splitting schemes are attributed to the transition from a complex problem to a chain of simpler problems. This book shows how to construct additive difference schemes (splitting schemes) to solve approximately unsteady multi-dimensional problems for PDEs. Two classes of schemes are highlighted: methods of splitting with respect to spatial variables (alternating direction methods) and schemes of splitting into physical processes. Also regionally additive schemes (domain decomposition methods) and unconditionally stable additive schemes of multi-component splitting are considered for evolutionary equations of first and second order as well as for systems of equations. The book is written for specialists in computational mathematics and mathematical modeling. All topics are presented in a clear and accessible manner.

Author : A.A. Samarskii,P.P. Matus,P.N. Vabishchevich
Publisher : Springer Science & Business Media
Page : 390 pages
File Size : 42,6 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401598743

Get Book

Difference Schemes with Operator Factors by A.A. Samarskii,P.P. Matus,P.N. Vabishchevich Pdf

Two-and three-level difference schemes for discretisation in time, in conjunction with finite difference or finite element approximations with respect to the space variables, are often used to solve numerically non stationary problems of mathematical physics. In the theoretical analysis of difference schemes our basic attention is paid to the problem of sta bility of a difference solution (or well posedness of a difference scheme) with respect to small perturbations of the initial conditions and the right hand side. The theory of stability of difference schemes develops in various di rections. The most important results on this subject can be found in the book by A.A. Samarskii and A.V. Goolin [Samarskii and Goolin, 1973]. The survey papers of V. Thomee [Thomee, 1969, Thomee, 1990], A.V. Goolin and A.A. Samarskii [Goolin and Samarskii, 1976], E. Tad more [Tadmor, 1987] should also be mentioned here. The stability theory is a basis for the analysis of the convergence of an approximative solu tion to the exact solution, provided that the mesh width tends to zero. In this case the required estimate for the truncation error follows from consideration of the corresponding problem for it and from a priori es timates of stability with respect to the initial data and the right hand side. Putting it briefly, this means the known result that consistency and stability imply convergence.

Author : Sergey Lemeshevsky,Piotr Matus,Dmitriy Poliakov
Publisher : Walter de Gruyter GmbH & Co KG
Page : 246 pages
File Size : 44,9 Mb
Release : 2016-09-26
Category : Mathematics
ISBN : 9783110489729

Get Book

Exact Finite-Difference Schemes by Sergey Lemeshevsky,Piotr Matus,Dmitriy Poliakov Pdf

Exact Finite-Difference Schemes is a first overview of the topic also describing the state-of-the-art in this field of numerical analysis. Construction of exact difference schemes for various parabolic and elliptic partial differential equations are discussed, including vibrations and transport problems. After this, applications are discussed, such as the discretisation of ODEs and PDEs and numerical methods for stochastic differential equations. Contents: Basic notation Preliminary results Hyperbolic equations Parabolic equations Use of exact difference schemes to construct NSFD discretizations of differential equations Exact and truncated difference schemes for boundary-value problem Exact difference schemes for stochastic differential equations Numerical blow-up time Bibliography

Author : Victor G. Ganzha,E. V. Vorozhtsov
Publisher : John Wiley & Sons
Page : 458 pages
File Size : 42,7 Mb
Release : 2011-03-01
Category : Science
ISBN : 9781118030851

Get Book

Computer-Aided Analysis of Difference Schemes for Partial Differential Equations by Victor G. Ganzha,E. V. Vorozhtsov Pdf

Advances in computer technology have conveniently coincided withtrends in numerical analysis toward increased complexity ofcomputational algorithms based on finite difference methods. It isno longer feasible to perform stability investigation of thesemethods manually--and no longer necessary. As this book shows,modern computer algebra tools can be combined with methods fromnumerical analysis to generate programs that will do the jobautomatically. Comprehensive, timely, and accessible--this is the definitivereference on the application of computerized symbolic manipulationsfor analyzing the stability of a wide range of difference schemes.In particular, it deals with those schemes that are used to solvecomplex physical problems in areas such as gas dynamics, heat andmass transfer, catastrophe theory, elasticity, shallow watertheory, and more. Introducing many new applications, methods, and concepts,Computer-Aided Analysis of Difference Schemes for PartialDifferential Equations * Shows how computational algebra expedites the task of stabilityanalysis--whatever the approach to stability investigation * Covers ten different approaches for each stability method * Deals with the specific characteristics of each method and itsapplication to problems commonly encountered by numerical modelers * Describes all basic mathematical formulas that are necessary toimplement each algorithm * Provides each formula in several global algebraic symboliclanguages, such as MAPLE, MATHEMATICA, and REDUCE * Includes numerous illustrations and thought-provoking examplesthroughout the text For mathematicians, physicists, and engineers, as well as forpostgraduate students, and for anyone involved with numericsolutions for real-world physical problems, this book provides avaluable resource, a helpful guide, and a head start ondevelopments for the twenty-first century.

Author : Ivan Gavrilyuk,Martin Hermann,Volodymyr Makarov,Myroslav V. Kutniv
Publisher : Springer Science & Business Media
Page : 247 pages
File Size : 52,8 Mb
Release : 2011-08-12
Category : Mathematics
ISBN : 9783034801072

Get Book

Exact and Truncated Difference Schemes for Boundary Value ODEs by Ivan Gavrilyuk,Martin Hermann,Volodymyr Makarov,Myroslav V. Kutniv Pdf

The book provides a comprehensive introduction to compact finite difference methods for solving boundary value ODEs with high accuracy. The corresponding theory is based on exact difference schemes (EDS) from which the implementable truncated difference schemes (TDS) are derived. The TDS are now competitive in terms of efficiency and accuracy with the well-studied numerical algorithms for the solution of initial value ODEs. Moreover, various a posteriori error estimators are presented which can be used in adaptive algorithms as important building blocks. The new class of EDS and TDS treated in this book can be considered as further developments of the results presented in the highly respected books of the Russian mathematician A. A. Samarskii. It is shown that the new Samarskii-like techniques open the horizon for the numerical treatment of more complicated problems. The book contains exercises and the corresponding solutions enabling the use as a course text or for self-study. Researchers and students from numerical methods, engineering and other sciences will find this book provides an accessible and self-contained introduction to numerical methods for solving boundary value ODEs.

Author : John C. Strikwerda
Publisher : SIAM
Page : 439 pages
File Size : 48,8 Mb
Release : 2007-09-20
Category : Mathematics
ISBN : 9780898716399

Get Book

Finite Difference Schemes and Partial Differential Equations by John C. Strikwerda Pdf

A unified and accessible introduction to the basic theory of finite difference schemes.

Author : Ronald E. Mickens
Publisher : World Scientific
Page : 268 pages
File Size : 43,8 Mb
Release : 2000
Category : Mathematics
ISBN : 981024133X

Get Book

Applications of Nonstandard Finite Difference Schemes by Ronald E. Mickens Pdf

The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter I gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations, that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used. This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.

Author : Ronald E Mickens
Publisher : World Scientific
Page : 664 pages
File Size : 50,6 Mb
Release : 2005-10-25
Category : Mathematics
ISBN : 9789814479868

Get Book

Advances in the Applications of Nonstandard Finite Difference Schemes by Ronald E Mickens Pdf

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic. Contents:Nonstandard Finite Difference Methods (R E Mickens)Application of Nonstandard Finite Difference Schemes to the Simulation Studies of Robotic Systems (R F Abo-Shanab et al.)Applications of Mickens Finite Differences to Several Related Boundary Value Problems (R Buckmire)High Accuracy Nonstandard Finite-Difference Time-Domain Algorithms for Computational Electromagnetics: Applications to Optics and Photonics (J B Cole)Nonstandard Finite Difference Schemes for Solving Nonlinear Micro Heat Transport Equations in Double-Layered Metal Thin Films Exposed to Ultrashort Pulsed Lasers (W Dai)Reliable Finite Difference Schemes with Applications in Mathematical Ecology (D T Dimitrov et al.)Applications of the Nonstandard Finite Difference Method in Non-Smooth Mechanics (Y Dumont)Finite Difference Schemes on Unbounded Domains (M Ehrhardt)Asymptotically Consistent Nonstandard Finite-Difference Methods for Solving Mathematical Models Arising in Population Biology (A B Gumel et al.)Nonstandard Finite Difference Methods and Biological Models (S R-J Jang)Robust Discretizations versus Increase of the Time Step for Chaotic Systems (C Letellier & E M A M Mendes)Contributions to the Theory of Nonstandard Finite-Difference Methods and Applications to Singular Perturbation Problems (J M-S Lubuma & K C Patidar)Frequency Accurate Finite Difference Methods (A L Perkins et al.)Nonstandard Discretization Methods on Lotka-Volterra Differential Equations (L-I W Roeger) Readership: Applied mathematicians, and researchers in numerical & computational mathematics and analysis & differential equations. Usable as a secondary text to a standard undergraduate or graduate course on numerical methods for differential equations. Keywords:Numerical Integration Methods;Finite Differences;Nonstandard Finite Difference Schemes;Differential Equations;Discrete Models;Numerical and Computational MathematicsKey Features:A collection of papers from renowned experts in their respective fieldsProvides the most recent work on the application of NSFD schemes and some of the mathematical analysis related to these schemes

Author : R. J. Sobey
Publisher : Unknown
Page : 50 pages
File Size : 55,5 Mb
Release : 1970
Category : Storm surges
ISBN : UOM:39015095032267

Get Book

Finite-difference Schemes Compared for Wave-deformation Characteristics in Mathematical Modeling of Two-dimensional Long-wave Propagation by R. J. Sobey Pdf

Author : A I Tolstykh
Publisher : World Scientific
Page : 332 pages
File Size : 48,5 Mb
Release : 1994-09-09
Category : Mathematics
ISBN : 9789814502351

Get Book

High Accuracy Non-Centered Compact Difference Schemes for Fluid Dynamics Applications by A I Tolstykh Pdf

This is the first book which describes completely the nontraditional difference schemes which combine the ideas of Padé-type approximation and upwind differencing. These possess some favorable properties and can be used to solve various problems in fluid dynamics and related disciplines. They were proposed by the author in the seventies and are extensively used in Russia. However, they seem to be relatively unknown outside the country. In this book, the author presents the theory of the schemes, to provide some sophisticated algorithms for different computational fluid dynamics problems, to supply readers with useful information which would permit them to construct a rich variety of algorithms of this type and to illustrate the applications of these methods to the numerical simulation of various fluid dynamics phenomena, ranging from supersonic viscous flows to some atmosphere and ocean processes. This book is an essential guide for anyone keenly interested in this field. Contents:IntroductionThird-Order Schemes with Compact Upwind DifferencingSome Extensions of Basic IdeasFifth-Order Non-Centered Compact SchemesHyperbolic SystemsCompact Upwind Schemes for Convection-Diffusion EquationsMultidimensional ProblemsCompressible Gas Flows Described by Navier-Stokes EquationsApplications to Incompressible Flow ProblemsA Solution-Dependent Coordinates for Grid GenerationSome Relevant Mathematical TopicsBibliographyIndex Readership: Applied mathematicians. keywords:High-Order Finite Difference Methods;Non-Centered Compact Differencing Operators;Upwind Compact Differencing;Upwind Compact Difference Schemes;For Hyperbolic Equations and Systems;For Compressible Navier Stokes Equations;For Incompressible Navier Stokes Equations;Primitive Variables Formulation Algorithms;Vorticity-Stream Function Formulation Algorithms;Solutions Procedures Relevant to Compact Schemes