There are so many excellent books on finite difference methods for ordinary and partial differential equations that writing yet another one requires a different view on the topic. The present book is not so concerned with the traditional academic presentation of the topic, but is focused at teaching the practitioner how to obtain reliable computations involving finite difference methods. This focus is based on a set of learning outcomes:
Various pedagogical elements are utilized to reach the learning outcomes, and these are commented upon next.
The book's overall pedagogical philosophy is the three-step process of first simplifying the problem to something we can understand in detail, and when that understanding is in place, we can generalize and hopefully address real-world applications with a sound scientific problem-solving approach. For example, in the chapter on a particular family of equations we first simplify the problem in question to a 1D, constant-coefficient equation with simple boundary conditions. We learn how to construct a finite difference method, how to implement it, and how to understand the behavior of the numerical solution. Then we can generalize to higher dimensions, variable coefficients, a source term, and more complicated boundary conditions. The solution of a compound problem is in this way an assembly of elements that are well understood in simpler settings.
This text favors a constructive approach to mathematics. Instead of a set of definitions followed by popping up a method, we emphasize how to think about the construction of a method. The aim is to obtain a good intuitive understanding of the mathematical methods.
The text is written in an easy-to-read style much inspired by the following quote.
Some people think that stiff challenges are the best device to induce learning, but I am not one of them. The natural way to learn something is by spending vast amounts of easy, enjoyable time at it. This goes whether you want to speak German, sight-read at the piano, type, or do mathematics. Give me the German storybook for fifth graders that I feel like reading in bed, not Goethe and a dictionary. The latter will bring rapid progress at first, then exhaustion and failure to resolve.The main thing to be said for stiff challenges is that inevitably we will encounter them, so we had better learn to face them boldly. Putting them in the curriculum can help teach us to do so. But for teaching the skill or subject matter itself, they are overrated. [1] (p. 86) Lloyd N. Trefethen, Applied Mathematician, 1955-.
This book assumes some basic knowledge of finite difference approximations, differential equations, and scientific Python or MATLAB programming, as often met in an introductory numerical methods course. Readers without this background may start with the light companion book "Finite Difference Computing with Exponential Decay Models" [2]. That book will in particular be a useful resource for the programming parts of the present book. Since the present book deals with partial differential equations, the reader is assumed to master multi-variable calculus and linear algebra.
Fundamental ideas and their associated scientific details are first introduced in the simplest possible differential equation setting, often an ordinary differential equation, but in a way that easily allows reuse in more complex settings with partial differential equations. With this approach, new concepts are introduced with a minimum of mathematical details. The text should therefore have a potential for use early in undergraduate student programs.
Many has experienced that "vast amounts of easy, enjoyable time", as stated in the quote above, arises when mathematics is implemented on a computer. The implementation process triggers understanding, creativity, and curiosity, but many students find the transition from a mathematical algorithm to a working code difficult and spend a lot of time on "programming issues".
Most books on numerical methods concentrate on the mathematics of the subject while details on going from the mathematics to a computer implementation are less in focus. A major purpose of this text is therefore to help the practitioner by providing all nuts and bolts necessary for safely going from the mathematics to a well-designed and well-tested computer code. A significant portion of the text is consequently devoted to programming details.
While MATLAB enjoys widespread popularity in books on numerical methods, we have chosen to use the Python programming language. Python is very similar to MATLAB, but contains a lot of modern software engineering tools that have become standard in the software industry and that should be adopted also for numerical computing projects. Python is at present also experiencing an exponential growth in popularity within the scientific computing community. One of the book's goals is to present an up-to-date Python eco system for implementing finite difference methods.
Program testing, called verification, is a key topic of the book. Good verification techniques are indispensable when debugging computer code, but also fundamental for achieving reliable simulations. Two verification techniques saturate the book: exact solution of discrete equations (where the approximation error vanishes) and empirical estimation of convergence rates in problems with exact (analytical or manufactured) solutions of the differential equation(s).
Traditional asymptotic analysis of errors is important for verification of code using convergence rates, but gives a limited understanding of how and why a correctly implemented numerical method may give non-physical results. By developing exact solutions, usually based on Fourier methods, of the discrete equations, one can obtain a physical understanding of the behavior of a numerical method. This approach is favored for analysis of methods in this book.
Our primary aim is to have a clean and easy-to-read computer code, and
we want a close one-to-one relationship between the computer code and
mathematical description of the algorithm. This principle calls for a
mathematical notation that is governed by the natural notation in the
computer code. The unknown is mostly called \( u \), but the meaning of
the symbol \( u \) in the mathematical description changes as we go from
the exact solution fulfilling the differential equation problem to the
symbol u that is naturally used in the code.
The aim of this book is not to give an overview of a lot of methods for a wide range of mathematical models. Such information can be found in numerous existing, more advanced books. The aim is rather to introduce basic concepts and a thorough understanding of how to think about computing with finite difference methods. We therefore go in depth with only the most fundamental methods and equations. However, we have a multi-disciplinary scope and address the interplay of mathematics, numerics, computer science, and physics.
Most book authors are careful with avoiding repetitions of material. The chapters in this book, however, contain some overlap, because I want the chapters to meaningful on their own. Modern publishing technology makes it easy to take selected chapters from different books to make a new book tailored to a specific course. The more a chapter builds on details in other chapters, the more difficult it is to reuse chapters in new contexts. Also, most readers find it convenient that the most important information is explicitly stated, even if it was already met in another chapter.
The author is particularly thankful to all the very detailed criticism of the text provided by Professor Svein Linge. Many students have provided lots of useful feedback on the exposition and found many errors in the text. Special efforts in this regard were made by Imran Ali, Shirin Fallahi, Anders Hafreager, Daniel Alexander Mo Søreide Houshmand, Kristian Gregorius Hustad, Mathilde Nygaard Kamperud, and Fatemeh Miri.