物理学计算入门：实战数值方法与参考书推荐

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Preface xv

value (Chapter 2) and boundary value form (Chapter 3). There are two chapters

on ﬁnite difference for PDEs: Chapter 4 covers static, linear operators (mainly

the Poisson problem), and in Chapter 5, we see time-dependent PDEs (both

linear, like Schr

odinger’s equation, and nonlinear forms). Chapter 6 develops

integration methods from interpolating polynomials, and connects the statistical

Monte Carlo integration method to simple box sums. Finally, I count the chapter

on the FFT as an integration topic.

2. Numerical linear algebra. Chapters 9–12 discuss numerical techniques for

matrix inversion, least squares solution, and the eigenvalue problem. Some

of the material in these chapters is required by topics in the ﬁrst section (for

example, solving Poisson’s problem in discretized form involves matrix inver-

sion) where we used canned routines. Now we come back and pick up these

topics, their motivation coming in part from previous discussions. Chapters 9

and 10 focus on “direct” solution of the matrix inverse and eigenvalue problem,

while Chapter 11 introduces iterative techniques (Jacobi, SOR, and Krylov sub-

space methods).

Chapter 12 is not, strictly speaking, a part of linear algebra –

but minimization is usefully applied in the nonlinear least squares problem, so

must come after the linear form of that problem.

3. Additional topics. The ﬁrst two sections provide methods that cover many phys-

ical problems. The remaining chapters give interesting additional topics, but

are slightly different in their coverage from the rest of the book. Chapter 13:

“Chaos,” for example, has no numerical methods attached to it. The neural net-

work chapter (14) presents these models in terms of their predictive ability for

physical phenomena, with emphasis on the promise provided by the Cybenko

theorem rather than the physiological inspiration normally encountered. Chap-

ter 15 returns to PDEs and describes the Galerkin approach with some additional

nonlinear PDE targets.

Two of the chapters do not fall neatly into these categories – the ﬁrst chapter provides

a review of the minimal programming tools that we need to use to implement

the methods presented in the other chapters – it also provides some orienting

Mathematica examples of these tools in action. The eighth chapter acts as a

pivot between the ﬁrst and second sections, and is meant to set up our interest in

the eigenvalue problem.

The grouping above is informed by the course that inspired this book, and for

which it was written – in it, the students present a ﬁnal project of their own design,

Divorcing the ﬁnite difference setup from the iterative solutions (Jacobi and SOR) to the matrix inverse problem

is not the traditional approach. Typically, authors will combine the discretization of the Laplacian with an

iterative update. But I believe that the two pieces are logically separate – discretizing the Laplace operator gives

us a matrix inverse problem to solve, how we go about solving it is a separate matter.

xvi Preface

and they begin working on this at about the middle of the semester. That is when we

have covered the bulk of item one above, so they have seen a number of different

kinds of problems and understand numerical methods for solving them. The section

on numerical linear algebra is interesting and informative, but students who need

to, for example, invert matrices for their ﬁnal projects can use a built-in routine

(few students have made the process of matrix inversion the central idea of their

projects) until the relevant material is covered (and even after, in most cases). The

additional topics section has then come at a time when students can take advantage

of the information in them without relying on it for their project (coupled, driven

oscillators can be solved and studied numerically before the nonlinear analysis

tools that are typically used to describe them are introduced).

While I use a subset of the programming language of Mathematica both in

the book, and in the course, I have found that students can re-render the content in

other languages relatively easily, and have had students work out the lab problems

using python, Java, C, Sage, and matlab. For those students with less of a pro-

gramming background, I provide, for each chapter, a Mathematica “notebook”

that contains all the commands used to generate the ﬁgures and example solu-

tions for the chapter. Then much of the programming can be done by re-working

examples from the chapter notebook. My intention is for students who are not as

interested in programming to have a way of learning the methods, without wor-

rying about their implementation as much. The computational methods are based

on implementation-independent mathematical ideas, and those are the core targets.

For students who enjoy the implementation side (I always did, although I was never

particularly good at it): optimizing and ordering commands efﬁciently, don’t look

at the chapter notebooks, work out your own routines from scratch.

This course is one of my favorites to teach – there are limitless physical problems

to set up, so it’s a great place for me to learn about new physics. Then the extent to

which it is difﬁcult to solve most problems (even simple ones) analytically is always

surprising. And, ﬁnally, the numerical solutions that allow progress are often simple

and fast. I usually cover each chapter in a week with three lectures: on Monday, we

just set up physical problems, ones that come from different physics, but all end

in a common, fundamental “problem.” On Wednesday, we develop a method that

solves that problem, and then, during “Friday potpourri,” we return to some of the

problems from Monday and solve them, and discuss limitations or extensions of

the method. The chapter structure mimics these lectures, with physical problems

presented ﬁrst, then methods, then additional items, sometimes additional physics,

other times, more in-depth discussion of the method or its variants.

What is notably missing from the weekly lectures is any sort of implementation

discussion – in addition to the three lectures, I also run three-hour “labs” every

week. Students come to these to work on the lab problems (I assign three or four

Preface xvii

of these each week), and this is where implementation can be carefully discussed,

with the relevant programming ideas built up on an individual basis, or through

group discussion. Having a concrete problem to work on (the lab problems) gives

natural focus to the programming. This separation between physics, the idea behind

the method, and its implementation, is intentional. The goal is to help identify

points of confusion as arising from the statement of a physical problem, the proper

functioning of a numerical method, or its implementation. Keeping those three

separate makes clearing up the confusion easier.

Website and materials

The book has a website, and on it, students can ﬁnd all of the chapter notebooks

(the Mathematica code that generated all the examples and ﬁgures in the book),

a few sample “lab notebooks,” and some project ideas that students have come up

with over the years. Some of the lab problems require mocked up data or other

information, and those are also available on the website.

Acknowledgements

It is a pleasure to thank my colleagues in the Reed College physics depart-

ment, especially Darrell Schroeter, who developed and taught an early version

of the computational methods class at Reed with me, Nelia Mann who con-

tributed useful comments and problems, David Latimer, a great sounding board,

and David Grifﬁths for his usual outstanding commentary, criticism and encour-

agement. Outside the department, Jim Fix gave me some great image rotation

advice, Olivia Schelly took the pictures that are used in Chapter 7, and Matt Sayre

wrote the Gravitea-Time song for ﬁltering. Simon Capelin and Lindsay Barnes at

Cambridge University Press have once again made the publishing process enjoy-

able, and I thank them for their help at all stages of the project.

I would like to thank Sebastian Doniach and the Stanford University physics

department for their hospitality in the winter of 2012: aside from a nice change of

scene, and stimulating new environment, I got to teach this material to an entirely

new audience, for which I am very grateful. The Writer’s Bloc has provided a

wonderful setting for revising the manuscript. Finally, a special thanks to the Reed

College students who helped me during the process of editing and reﬁning the

book: Tom Chartrand, Todd Garon, and Reuven Lazarus.

Programming overview

A programming language useful to this book must provide a minimal set of compo-

nents that can be used to combine numbers, compare quantities and act on the result

of that comparison, repeat operations until a condition is met, and contain functions

that we can use to input data, and output results. Almost any language will sufﬁce,

but I have chosen to use Mathematica’s programming environment as the vehi-

cle. The reasoning is that 1. The input/output functions of Mathematica are easy

to use, and require little additional preparation

2. We will be focused on the ideas,

numerical and otherwise, associated with the methods we study, and I want to draw

a clear distinction between those ideas and issues of implementation. This book is

not meant to teach you everything you need to know about programming

– we will

discuss only the bare essentials needed to implement the methods. Instead, we will

focus on the physical motivation and tools of analysis for a variety of techniques.

My hope is that the use of Mathematica allows us to discuss implementation

in a homogeneous way, and our restriction to the basic programming structure

of Mathematica (as opposed to the higher-level functionality) allows for easy

porting to the language of your choice.

Here, we will review the basic operations, rendered in Mathematica, falling

into the broad categories: arithmetic operations, comparisons, loops, and input–

output routines. In addition, we must be able to use variable names that can be

assigned values, and there is a scoping for these constructions in Mathematica

similar to C (and many other languages). Functions, in the sense of C, exist in

Mathematica, and we will use a particular (safe) form, although depending

on the context, there are faster (and slower) ways to generate functions. We will

bundle almost every set of computations into a function, and this is to mimic

There are libraries to import audio and video, for example, in C, but the resulting internal representation can

be difﬁcult to work with. The details of compiling with those libraries correctly linked is also speciﬁc to the

language and compiler, issues that I want to avoid.

Although, I will employ good programming practice in the examples and accompanying chapter notebooks.

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