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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A comprehensive guide to numerical methods for simulating physical-chemical systems This book offers a systematic, highly accessible presentation of numerical methods used to simulate the behavior of physical-chemical systems. Unlike most books on the subject, it focuses on methodology rather than specific applications. Written for students and professionals across an array of scientific and engineering disciplines and with varying levels of experience with applied mathematics, it provides comprehensive descriptions of numerical methods without requiring an advanced mathematical background. Based on its author's more than forty years of experience teaching numerical methods to engineering students, Numerical Methods for Solving Partial Differential Equations presents the fundamentals of all of the commonly used numerical methods for solving differential equations at a level appropriate for advanced undergraduates and first-year graduate students in science and engineering. Throughout, elementary examples show how numerical methods are used to solve generic versions of equations that arise in many scientific and engineering disciplines. In writing it, the author took pains to ensure that no assumptions were made about the background discipline of the reader. * Covers the spectrum of numerical methods that are used to simulate the behavior of physical-chemical systems that occur in science and engineering * Written by a professor of engineering with more than forty years of experience teaching numerical methods to engineers * Requires only elementary knowledge of differential equations and matrix algebra to master the material * Designed to teach students to understand, appreciate and apply the basic mathematics and equations on which Mathcad and similar commercial software packages are based Comprehensive yet accessible to readers with limited mathematical knowledge, Numerical Methods for Solving Partial Differential Equations is an excellent text for advanced undergraduates and first-year graduate students in the sciences and engineering. It is also a valuable working reference for professionals in engineering, physics, chemistry, computer science, and applied mathematics.
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Veröffentlichungsjahr: 2018
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Numerical Methods for Solving Partial Differential Equations
A Comprehensive Introduction for Scientists and Engineers
George F. Pinder
This edition first published 2018 by John Wiley and Sons, Inc.
© 2018 by John Wiley & Sons, Inc.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of George F Pinder to be identified as the author of this work has been asserted in accordance with law.
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Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging-in-Publication Data:
Hardback ISBN: 978-1-119-31611-4
Cover image by Charles Bombard Cover design by Wiley
Robert N. Farvolden and John D. Bredehoeft my mentors
Contents
Preface
CHAPTER 1: Interpolation
1.1 Purpose
1.2 Definitions
1.3 Example
1.4 Weirstraus Approximation Theorem
1.5 Lagrange Interpolation
1.6 Compare
P
2
(
θ
) and
f
ˆ
(
θ
)
1.7 Error of Approximation
1.8 Multiple Elements
1.9 Hermite Polynomials
1.10 Error in Approximation by Hermites
1.11 Chapter Summary
1.12 Problems
Bibliography
CHAPTER 2: Numerical Differentiation
2.1 General Theory
2.2 Two-point Difference Formulae
2.3 Two-point Formulae from Taylor Series
2.4 Three-point Difference Formulae
2.5 Chapter Summary
2.6 Problems
Notes
Bibliography
CHAPTER 3: Numerical Integration
3.1 Newton-Cotes Quadrature Formulas
3.2 Chapter Summary
3.3 Problems
Bibliography
CHAPTER 4: Initial Value Problems
4.1 Euler Forward Int egrat ion Method Example
4.2 Convergence
4.3 Consistency
4.4 Stability
4.5 Lax Equivalence Theorem
4.6 Runge-Kutta Type Formulas
4.7 Chapter Summary
4.8 Problems
Notes
Bibliography
CHAPTER 5: Weighted Residuals Methods
5.1 Finite Volume or Subdomain Method
5.2 Galerkin Method for First Order Equations
5.3 Galerkin Method for Second Order Equations
5.4 Finite Volume Method for Second-Order Equations
5.5 Collocation Method
5.6 Chapter Summary
5.7 Problems
Bibliography
CHAPTER 6: Initial Boundary-Value Problems
6.1 Introduction
6.2 Two Dimensional Polynomial Approximations
6.3 Finite Difference Approximation
6.4 Stability of Finite Difference Approximations
6.5 Galerkin Finite Element Approximations in Time
6.6 Chapter Summary
6.7 Problems
Bibliography
CHAPTER 7: Finite Difference Methods in Two Space
7.1 Example Problem
7.2 Chapter Summary
7.3 Problems
Bibliography
CHAPTER 8: Finite Element Methods in Two Space
8.1 Finite Element Approximations over Rectangles
8.2 Finite Element Approximations over Triangles
8.3 Isoparametric Finite Element Approximation
8.4 Chapter Summary
8.5 Problems
Bibliography
CHAPTER 9: Finite Volume Approximation in Two Space
9.1 Finite Volume Formulation
9.2 Finite Volume Example Problem 1
9.3 Finite Volume Example Problem Two
9.4 Chapter Summary
9.5 Problems
Bibliography
CHAPTER 10: Initial Boundary-value Problems
10.1 Mass Lumping
10.2 Chapter Summary
10.3 Problems
Bibliography
CHAPTER 11: Boundary-Value Problems in Three Space
11.1 Finite Difference Approximations
11.2 Finite Element Approximations
11.3 Chapter Summary
Bibliography
CHAPTER 12: Nomenclature
Index
EULA
List of Tables
Table 1.1
Table 1.2
Table 1.3
Table 1.4
Table 1.5
Table 1.6
Table 9.1
List of Illustrations
Figure 1.1
Figure 1.2
Figure 1.3
Figure 1.4
Figure 1.5
Figure 1.6
Figure 1.7
Figure 1.8
Figure 1.9
Figure 1.10
Figure 1.11
Figure 1.12
Figure 1.13
Figure 1.14
Figure 1.15
Figure 1.16
Figure 1.17
Figure 1.18
Figure 1.19
Figure 1.20
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 3.1
Figure 4.1
Figure 4.2
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15
Figure 5.16
Figure 5.17
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 6.10
Figure 6.11
Figure 6.12
Figure 6.13
Figure 6.14
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
Figure 7.7
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Figure 8.6
Figure 8.7
Figure 8.8
Figure 8.9
Figure 8.10
Figure 8.11
Figure 8.12
Figure 8.13
Figure 8.14
Figure 8.15
Figure 8.17
Figure 8.18
Figure 8.19
Figure 8.20
Figure 8.21
Figure 8.22
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Figure 9.5
Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9
Figure 9.10
Figure 11.1
Figure 11.3
Guide
Cover
Table of Contents
Chapter 1
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Preface
While there are many good books on numerical methods suitable for students of mathematics and many others that are accessible to scientists and engineers, but dedicated to a specific discipline, there is a need for a book that is accessible to students of science and engineering that is not discipline specific, yet rigorous and comprehensive in scope. This book is an effort to fill this need.
Herein I provide the logical underpinnings of all of the commonly encountered numerical methods, namely finite difference, finite element, collocation, and finite volume methods, at a level of sophistication consistent with the needs and interests of science and engineering students. Two mathematical concepts, namely polynomial approximation theory and the method of weighted residuals, form the intellectual framework for the introduction and explanation of all of these methods.
The approach is to first introduce polynomial approximation theory in one space dimension followed by the concept of the methods of weighted residuals. Employing only polynomial approximation theory the finite difference method is easily developed and presented. With the addition of the method of weighted residuals, finite element, collocation and finite volume methods are readily accessible. These concepts are introduced first in one space dimension, then the time dimension, then two space dimensions, and finally two space dimensions and time.
The equations considered are first order, second order, and second order in space and first order in time. By design, the book does not focus on any specific area of science or engineering. It is designed to teach numerical methods as a concept rather than as applied to a specific discipline. The intent is to provide the student with the ability to understand numerical methods as encountered in technical readings specific to his/her discipline and to be able to apply them in practice.
The book assumes a knowledge of matrix algebra and differential equations. A programming language is also needed if the reader is interested in applying numerical methods to example problems. No prior knowledge of numerical methods is assumed. While a few theorems are used, no proofs are presented.
This book stems from a course I teach in Numerical Methods for Engineers. The course is taught as a precept and typically populated by an approximately equal number of senior undergraduates and graduate students from different engineering disciplines. A project of practical significance is assigned that requires the creation of a computer program capable of solving a second-order two-space dimensional equation using finite elements.
I am indebted to Xin Kou, my doctoral student in mathematics, for carefully reviewing the manuscript for his book, identifying notational inconsistencies and making important suggestions as to how to improve the presentation.
