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
DOWNLOAD EBOOKThis 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.
New Difference Schemes for Partial Differential Equations
Author: Allaberen Ashyralyev
Publisher: Birkhäuser
Published: 2012-12-06
Total Pages: 453
ISBN-13: 3034879229
DOWNLOAD EBOOKThis 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.
The Theory of Difference Schemes
Author: Alexander A. Samarskii
Publisher: CRC Press
Published: 2001-03-29
Total Pages: 796
ISBN-13: 9780203908518
DOWNLOAD EBOOKThe 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."
Difference Schemes
Author: S.K. Godunov
Publisher: Elsevier
Published: 1987-05-01
Total Pages: 509
ISBN-13: 0080875408
DOWNLOAD EBOOKMuch 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.
Applications of Nonstandard Finite Difference Schemes
Author: Ronald E. Mickens
Publisher: World Scientific
Published: 2000
Total Pages: 268
ISBN-13: 9789810241339
DOWNLOAD EBOOKThe 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.
Nonstandard Finite Difference Schemes: Methodology And Applications
Author: Ronald E Mickens
Publisher: World Scientific
Published: 2020-11-11
Total Pages: 332
ISBN-13: 981122255X
DOWNLOAD EBOOKThis 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.
Additive Operator-Difference Schemes
Author: Petr N. Vabishchevich
Publisher: Walter de Gruyter
Published: 2013-11-27
Total Pages: 370
ISBN-13: 3110321467
DOWNLOAD EBOOKApplied 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.
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
DOWNLOAD EBOOKThis 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.
Nonstandard Finite Difference Models of Differential Equations
Author: Ronald E. Mickens
Publisher: World Scientific
Published: 1994
Total Pages: 264
ISBN-13: 9810214588
DOWNLOAD EBOOKThis 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.
