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The Theory of Difference Schemes

Author: Alexander A. Samarskii

Publisher: CRC Press

Published: 2001-03-29

Total Pages: 796

ISBN-13: 9780203908518

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."

The Theory of Difference Schemes

Author: Alexander A. Samarskii

Publisher: CRC Press

Published: 2001-03-29

Total Pages: 788

ISBN-13: 0203908511

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New Difference Schemes for Partial Differential Equations

Author: Allaberen Ashyralyev

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 453

ISBN-13: 3034879229

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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.

Finite Difference Schemes and Partial Differential Equations

Author: John C. Strikwerda

Publisher: Springer

Published: 1989-09-28

Total Pages: 410

ISBN-13:

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Analysis of Finite Difference Schemes

Author: Boško S. Jovanović

Publisher: Springer Science & Business Media

Published: 2013-10-22

Total Pages: 416

ISBN-13: 1447154606

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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.

Finite Difference Methods for Ordinary and Partial Differential Equations

Author: Randall J. LeVeque

Publisher: SIAM

Published: 2007-01-01

Total Pages: 356

ISBN-13: 9780898717839

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This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Difference Equations, Second Edition

Author: R Mickens

Publisher: CRC Press

Published: 1991-01-01

Total Pages: 470

ISBN-13: 9780442001360

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In recent years, the study of difference equations has acquired a new significance, due in large part to their use in the formulation and analysis of discrete-time systems, the numerical integration of differential equations by finite-difference schemes, and the study of deterministic chaos. The second edition of Difference Equations: Theory and Applications provides a thorough listing of all major theorems along with proofs. The text treats the case of first-order difference equations in detail, using both analytical and geometrical methods. Both ordinary and partial difference equations are considered, along with a variety of special nonlinear forms for which exact solutions can be determined. Numerous worked examples and problems allow readers to fully understand the material in the text. They also give possible generalization of the theorems and application models. The text's expanded coverage of application helps readers appreciate the benefits of using difference equations in the modeling and analysis of "realistic" problems from a broad range of fields. The second edition presents, analyzes, and discusses a large number of applications from the mathematical, biological, physical, and social sciences. Discussions on perturbation methods and difference equation models of differential equation models of differential equations represent contributions by the author to the research literature. Reference to original literature show how the elementary models of the book can be extended to more realistic situations. Difference Equations, Second Edition gives readers a background in discrete mathematics that many workers in science-oriented industries need as part of their general scientific knowledge. With its minimal mathematical background requirements of general algebra and calculus, this unique volume will be used extensively by students and professional in science and technology, in areas such as applied mathematics, control theory, population science, economics, and electronic circuits, especially discrete signal processing.

Nonstandard Finite Difference Schemes: Methodology And Applications

Author: Ronald E Mickens

Publisher: World Scientific

Published: 2020-11-11

Total Pages: 332

ISBN-13: 981122255X

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This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.

High Accuracy Non-Centered Compact Difference Schemes for Fluid Dynamics Applications

Author: A I Tolstykh

Publisher: World Scientific

Published: 1994-09-09

Total Pages: 332

ISBN-13: 9814502359

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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

High Order Difference Methods for Time Dependent PDE

Author: Bertil Gustafsson

Publisher: Springer Science & Business Media

Published: 2007-12-06

Total Pages: 343

ISBN-13: 3540749934

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This book covers high order finite difference methods for time dependent PDE. It gives an overview of the basic theory and construction principles by using model examples. The book also contains a general presentation of the techniques and results for well-posedness and stability, with inclusion of the three fundamental methods of analysis both for PDE in its original and discretized form: the Fourier transform, the eneregy method and the Laplace transform.