Analysis of Finite Difference Schemes

Analysis of Finite Difference Schemes
Title Analysis of Finite Difference Schemes PDF eBook
Author Boško S. Jovanović
Publisher Springer Science & Business Media
Pages 416
Release 2013-10-22
Genre Mathematics
ISBN 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

Finite Difference Methods for Ordinary and Partial Differential Equations
Title Finite Difference Methods for Ordinary and Partial Differential Equations PDF eBook
Author Randall J. LeVeque
Publisher SIAM
Pages 356
Release 2007-01-01
Genre Mathematics
ISBN 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.

Finite Difference Schemes and Partial Differential Equations

Finite Difference Schemes and Partial Differential Equations
Title Finite Difference Schemes and Partial Differential Equations PDF eBook
Author John C. Strikwerda
Publisher Springer
Pages 410
Release 1989-09-28
Genre Juvenile Nonfiction
ISBN

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Applications of Nonstandard Finite Difference Schemes

Applications of Nonstandard Finite Difference Schemes
Title Applications of Nonstandard Finite Difference Schemes PDF eBook
Author Ronald E. Mickens
Publisher World Scientific
Pages 268
Release 2000
Genre Mathematics
ISBN 9789810241339

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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.

Finite Difference Computing with PDEs

Finite Difference Computing with PDEs
Title Finite Difference Computing with PDEs PDF eBook
Author Hans Petter Langtangen
Publisher Springer
Pages 522
Release 2017-06-21
Genre Computers
ISBN 3319554565

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This book is open access under a CC BY 4.0 license. This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.

Finite Difference Methods on Irregular Networks

Finite Difference Methods on Irregular Networks
Title Finite Difference Methods on Irregular Networks PDF eBook
Author Bernd Heinrich
Publisher Walter de Gruyter GmbH & Co KG
Pages 212
Release 1987-12-31
Genre Mathematics
ISBN 311272089X

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No detailed description available for "Finite Difference Methods on Irregular Networks".

Nonstandard Finite Difference Models of Differential Equations

Nonstandard Finite Difference Models of Differential Equations
Title Nonstandard Finite Difference Models of Differential Equations PDF eBook
Author Ronald E. Mickens
Publisher World Scientific
Pages 264
Release 1994
Genre Mathematics
ISBN 9810214588

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This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of ?exact? and ?best? finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation.