George F. Pinder - Numerical Methods for Solving Partial Differential Equations: A Comprehensive Introduction for Scientists and Engineers

This edition first published 2018 by John Wiley and Sons, Inc.

2018 by John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication Data:

Hardback ISBN: 978-1-119-31611-4

Cover image by Charles Bombard
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Robert N. Farvolden and John D. Bredehoeft my mentors

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.

In this chapter, we will introduce interpolation theory, the first of two key topics that will form the foundation of everything that follows in this book. We will find that this concept leads quite naturally to finite difference methods and, when combined with the second key topic, the method of weighted residuals, provides the necessary mathematical concepts needed for all other numerical methods we present. So, let us get started.

Interpolation is a method of constructing new data points between known values or for creating a function that fits exactly a known set of discrete data points defined within a specific range. Interpolation has many applications in science and engineering. In this book, it will be used to form the basis for numerical differentiation, numerical quadrature, numerical integration, and as a key part of several numerical methods used to solve differential and partial differential equations.

We begin by introducing some interpolation notation. Consider a region

(1.1)

as illustrated in

Discretized line spanning a to b.

Next assume there exists a function f(x) that is a known function of