Python编程：科学计算与数值模拟入门

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"Programming for Computations - Python" 是一本针对初学者的编程教材，专注于使用Python语言解决数学问题。本书的作者是Svein Linge和Hans Petter Langtangen，他们通过简洁易懂的方式介绍了科学计算的基础，旨在帮助没有编程经验的学生快速掌握编写程序来解决工程和科学课程中的数值方法问题。书中强调了通用算法、程序设计的清晰性、函数的使用以及自动测试来验证程序正确性的方法。书的内容分为六个章节，涵盖了以下几个关键知识点： 1. **The First Few Steps**：这部分介绍了编程的基础，包括Python环境的设置，基本的数据类型（如整数、浮点数、字符串和布尔值），以及输入和输出操作。 2. **Basic Constructions**：这部分深入到控制流程，讲解了条件语句（if-else）和循环（for和while），这些都是编写任何程序的基础。此外，还介绍了函数的定义和调用，以及如何组织代码以提高可读性和可维护性。 3. **Computing Integrals**：在这一章，作者讲解了如何使用Python进行数值积分，这是许多物理和工程问题中的常见计算。这涉及到数值方法，如梯形法则、辛普森法则等。 4. **Solving Ordinary Differential Equations**：这部分介绍了解常微分方程（ODEs）的数值方法，如欧拉方法、龙格-库塔方法，这对于模拟动态系统至关重要。 5. **Solving Partial Differential Equations**：这部分扩展到偏微分方程（PDEs）的数值解，可能涉及有限差分法或有限元方法，这些方法广泛应用于流体动力学、热传导等领域。 6. 另外，书中还可能涵盖错误和异常处理、数据可视化、文件操作等主题，这些都是进行科学计算时不可或缺的部分。这本书属于 Springer 的 "Texts in Computational Science and Engineering" 系列，适合工程和科学专业的学生，以及对使用Python进行数值计算感兴趣的读者。书中所提及的数学分类（2010年版）包括实分析、常微分方程、数值方法等领域，旨在通过编程实现对这些理论知识的实践应用。通过阅读这本书，读者将获得编写Python程序来解决复杂数学问题的能力，这对于现代科学研究和工程实践至关重要。书中的实例和练习将帮助读者逐步建立对Python编程和数值方法的理解，进而能够进行简单的数值模拟和计算。

xvi List of Exercises

Exercise 3.6: Explore rounding errors with large numbers ............ 89

Exercise 3.7: Write test functions for

xdx ................... 89

Exercise 3.8: Rectangle methods ............................ 89

Exercise 3.9: Adaptive integration ........................... 90

Exercise 3.10: Integrating x raised to x ........................ 90

Exercise 3.11: Integrate products of sine functions ................. 91

Exercise 3.12: Revisit ﬁt of sines to a function ................... 91

Exercise 3.13: Derive the trapezoidal rule for a double integral ......... 92

Exercise 3.14: Compute the area of a triangle by Monte Carlo integration ... 92

Exercise 4.1: Geometric construction of the Forward Euler method ....... 153

Exercise 4.2: Make test functions for the Forward Euler method ........ 153

Exercise 4.3: Implement and evaluate Heun’s method ............... 153

Exercise 4.4: Find an appropriate time step; logistic model ............ 154

Exercise 4.5: Find an appropriate time step; SIR model .............. 154

Exercise 4.6: Model an adaptive vaccination campaign .............. 154

Exercise 4.7: Make a SIRV model with time-limited effect of vaccination ... 154

Exercise 4.8: Refactor a ﬂat program ......................... 155

Exercise 4.9: Simulate oscillations by a general ODE solver ........... 155

Exercise 4.10: Compute the energy in oscillations ................. 155

Exercise 4.11: Use a Backward Euler scheme for population growth ...... 156

Exercise 4.12: Use a Crank-Nicolson scheme for population growth ...... 156

Exercise 4.13: Understand ﬁnite differences via Taylor series .......... 157

Exercise 4.14: Use a Backward Euler scheme for oscillations .......... 158

Exercise 4.15: Use Heun’s method for the SIR model ............... 159

Exercise 4.16: Use Odespy to solve a simple ODE ................. 159

Exercise 4.17: Set up a Backward Euler scheme for oscillations ......... 159

Exercise 4.18: Set up a Forward Euler scheme for nonlinear and damped

oscillations ............................... 160

Exercise 4.19: Discretize an initial condition ..................... 160

Exercise 5.1: Simulate a diffusion equation by hand ................ 177

Exercise 5.2: Compute temperature variations in the ground ........... 178

Exercise 5.3: Compare implicit methods ....................... 178

Exercise 5.4: Explore adaptive and implicit methods ................ 179

Exercise 5.5: Investigate the  rule ........................... 179

Exercise 5.6: Compute the diffusion of a Gaussian peak

.............. 180

Exercise 5.7: Vectorize a function for computing the area of a polygon .... 181

Exercise 5.8: Explore symmetry ............................ 181

Exercise 5.9: Compute solutions as t !1 ..................... 182

Exercise 5.10: Solve a two-point boundary value problem ............ 183

Exercise 6.1: Understand why Newton’s method can fail ............. 206

Exercise 6.2: See if the secant method fails ..................... 207

Exercise 6.3: Understand why the bisection method cannot fail ......... 207

Exercise 6.4: Combine the bisection method with Newton’s method ...... 207

Exercise 6.5: Write a test function for Newton’s method ............. 207

Exercise 6.6: Solve nonlinear equation for a vibrating beam ........... 207

21TheFirstFewSteps

may want to analyze these data in Excel and make some graphics out of it. How-

ever, assume there is no menu in Excel that allows you to import data in this speciﬁc

format. Excel can work with many different data formats, but not this one. You start

searching for alternatives to Excel that can do the same and read this type of data

ﬁles. Maybe you cannot ﬁnd any ready-made program directly applicable. You

have reached the point where knowing how to write programs on your own would

be of great help to you! With some programming skills, you may write your own

little program which can translate one data format to another. With that little piece

of tailored code, your data may be read and analyzed, perhaps in Excel, or perhaps

by a new program tailored to the computations that the measurement data demand.

The real power of computers can only be utilized if you can program them.

By programming you can get the computer to do (most often!) exactly what you

want. Programming consists of writing a set of instructions in a very specialized

language that has adopted words and expressions from English. Such languages

are known as programming or computer languages. The set of instructions is given

to a program which can translate the meaning of the instructions into real actions

inside the computer.

The purpose of this book is to teach you to write such instructions dedicated to

solve mathematical and engineering problems by fundamental numerical methods.

There are numerous computer languages for different purposes. Within the en-

gineering area, the most widely used computer languages are Python, MATLAB,

Octave, Fortran, C, C++, and to some extent Maple, and Mathematica. How you

write the instructions (i.e. the syntax) differs between the languages. Let us use an

analogy.

Assume you are an international kind of person, having friends abroad in Eng-

land, Russia and China. They want to try your favorite cake. What can you do?

Well, you may write down the recipe in those three languages and send them over.

Now, if you have been able to think correctly when writing down the recipe, and

you have written the explanations according to the rules in each language, each of

your friends will produce the same cake. Your recipe is the “computer program”,

while English, Russian and Chinese represent the “computer languages” with their

own rules of how to write things. The end product, though, is still the same cake.

Note that you may unintentionally introduce errors in your “recipe”. Depending on

the error, this may cause “baking execution” to stop, or perhaps produce the wrong

cake. In your computer program, the errors you introduce are called bugs (yes,

small insects! . . . for historical reasons), and the process of ﬁxing them is called

debugging. When you try to run your program that contains errors, you usually

get warnings or error messages. However, the response you get depends on the er-

ror and the programming language. You may even get no response, but simply the

wrong “cake”. Note that the rules of a programming language have to be followed

very strictly. This differs from languages like English etc., where the meaning might

be understood even with spelling errors and “slang” included.

This book comes in two versions, one that is based on Python, and one based on

Matlab. Both Python and Matlab represent excellent programming environments

for scientiﬁc and engineering tasks. The version you are reading now, is the Python

version.

Some of Python’s strong properties deserve mention here: Many global func-

tions can be placed in only one ﬁle, functions are straightforwardly transferred as

1.2 A Python Program with Variables 3

arguments to other functions, there is good support for interfacing C, C++ and For-

tran code (i.e., a Python program may use code written in other languages), and

functions explicitly written for scalar input often work ﬁne (without modiﬁcation)

also with vector input. Another important thing, is that Python is available for free.

It can be downloaded from the Internet an d will run on most platforms.

Readers who want to expand their scientiﬁc programming skills beyond the

introductory level of the present exposition, are encouraged to study the book

A Primer on Scientiﬁc Programming with Python [13]. This comprehensive book

is as suitable for beginners as for professional programmers, and teaches the art

of programming through a huge collection of dedicated examples. This book is

considered the primary reference, and a natural extension, of the programming

matters in the present book.

Some computer science terms

Note that, quite often, the terms script and scripting are used as synonyms for

program and programming, respectively.

The inventor of the Perl programming language, Larry Wall, tried to explain

the difference between script and program in a humorous way (from perl.com

Suppose you went back to Ada Lovelace

and asked her the difference between

a script and a program. She’d probably look at you funny, then say something

like: Well, a script is what you give the actors, but a program is what you give

the audience. That Ada was one sharp lady . . . Since her time, we seem to

have gotten a bit more confused about what we mean when we say scripting. It

confuses even me, and I’m supposed to be one of the experts.

There are many other widely used computer science terms to pick up. Writing

a program (or script or code) is often expressed as implementing the program.

Executing a program means running the program. An algorithm is a recipe for

how to construct a program. A bug is an error in a program, and the art of

tracking down and removing bugs is called debugging. Simulating or simulation

refers to using a program to mimic processes in the real world, often through

solving differential equations that govern the physics of the processes.

1.2 A Python Program with Variables

Our ﬁrst example regards programming a mathematical model that predicts the po-

sition of a ball thrown up in the air. From Newton’s 2nd law, and by assuming

negligible air resistance, one can derive a mathematical model that predicts the ver-

tical position y of the ball at time t. From the model one gets the formula

y D v

t  0:5gt

;

where v

is the initial upwards velocity and g is the acceleration of gravity, for

which 9:81 ms

2

is a reasonable value (even if it depends on things like location

on the earth). With this formula at hand, and when v

is known, you may plug in

a value for time and get out the corresponding height.

http://www.perl.com/pub/2007/12/06/soto-11.html

http://en.wikipedia.org/wiki/Ada_Lovelace

41TheFirstFewSteps

1.2.1 The Program

Let us next look at a Python program for evaluating this simple formula. Assume

the program is contained as text in a ﬁle named ball.py. The text looks as follows

(ﬁle ball.py):

# Program for computing the height of a ball in vertical motion

v0 =5 # Initial velocity

g = 9.81 # Acceleration of gravity

t = 0.6 # Time

y = v0*t - 0.5*g*t**2 # Vertical position

print y

Computer programs and parts of programs are typeset with a blue background

in this book. A slightly darker top and bottom bar, as above, indicates that the code

is a complete program that can be run as it stands. Without the bars, the code is just

a snippet and will (normally) need additional lines to run properly.

1.2.2 Dissection of the Program

A computer program is plain text, as here in the ﬁle ball.py, which contains in-

structions to the computer. Humans can read the code and understand what the

program is capable of doing, but the program itself does not trigger any actions on

a computer before another program, the Python interpreter, reads the program text

and translates this text into speciﬁc actions.

You must learn to play the role of a computer

Although Python is responsible for reading and understanding your program, it is

of fundamental importance that you fully understand the program yourself. You

have to know the implication of every instruction in the program and be able to

ﬁgure out the consequences of the instructions. In other words, you must be able

to play the role of a computer. The reason for this strong demand of knowledge is

that errors unavoidably, and quite often, will be committed in the program text,

and to track down these errors, you have to simulate what the computer does

with the program. Next, we shall explain all the text in

ball.py in full detail.

When you run your program in Python, it will interpret the text in your ﬁle line

by line, from the top, reading each line from left to right. The ﬁrst line it reads is

# Program for computing the height of a ball in vertical motion.

This line is what we call a comment. That is, the line is not meant for Python to read

and execute, but rather for a human that reads the code and tries to understand what

is going on. Therefore, one rule in Python says that whenever Python encounters

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Python编程：科学计算与数值模拟入门

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