Computational physiCs

351 Pages·2022·5.74 MB·English

by  Darren J. Walker

C
omputational
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hysiCs
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C
omputational
p
hysiCs
s e
eCond dition
D J. W
arren alker
MERCURY LEARNING AND INFORMATION
Dulles, Virginia
Boston, Massachusetts
New Delhi
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Copyright ©2022 by Mercury Learning and inforMation LLC. All rights reserved.
Original title and copyright: Computational Physics: An Undergraduate’s Guide,
2/E. Copyright ©2021 by D.J. Walker. All rights reserved. Published by Pantaneto
Press.
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D. J. Walker. Computational Physics, Second Edition.
ISBN: 978-1-68392-832-4
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To Charlotte
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C
ontents
Chapter 1: Introduction 1
1.1 Getting Started with Coding 1
1.2 Getting To Know The Linux Command Line 3
1.3 Bonjour Tout Le Monde 6
1.4 The Rest of the Book 12
Chapter 2: Getting Comfortable 15
2.1 Computers: What You Should Know 15
2.1.1 Hardware 15
2.1.2 Software 17
2.1.3 Number Representation and
Precision 19
2.2 Some Important Mathematics 24
2.2.1 Taylor Series 25
2.2.2 Matrices: A Brief Overview 27
Exercises 32
Chapter 3: Interpolation and Data Fitting 35
3.1 Interpolation 35
3.1.1 Linear Interpolation 35
3.1.2 Polynomial Interpolation 38
3.1.2 Cubic Spline 43
3.2 Data Fitting 45
3.2.1 Regression: Illustrative Example 45
3.2.2 Linear Least Squares: Matrix Form 48
3.2.3 Realistic Example: Millikan’s
Experiment 50
Exercises 53
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viii • Contents
Chapter 4: Searching for Roots 55
4.1 Finding Roots 55
4.1.1 Bisection 56
4.1.2 Newton–Raphson 58
4.1.3 Secant 60
4.2 Hybrid Methods 62
4.2.1 Bisection–Newton–Raphson 62
4.2.2 Brute Force Search 64
4.3 What’s The Point of Root Searching? 65
4.3.1 The Infinite Square Well 65
4.3.2 The Finite Square Well 69
4.3.3 Programming the Root Finder 72
Exercises 77
Chapter 5: Numerical Quadrature 81
5.1 Simple Quadrature 82
5.1.1 The Mid-Ordinate Rule 82
5.1.2 The Trapezoidal Rule 83
5.1.3 Simpson’s Rule 84
5.2. Advanced Quadrature 85
5.2.1 Euler–Maclaurin Integration 85
5.2.2 Adaptive Quadrature 86
5.2.3 Multidimensional Integration 90
Exercises 93
Chapter 6: Ordinary Differential Equations 95
6.1 Classification of Differential Equations 96
6.1.1 Types of Differential Equations 96
6.1.2 Types of Solution and Initial
Conditions 98
6.2 Solving First-Order ODEs 99
6.2.1 Simple Euler Method 99
6.2.2 Modified and Improved Euler
Methods 102
6.2.3 The Runge–Kutta Method 104
6.2.4 Adaptive Runge–Kutta 107
6.3 Solving Second-Ordered ODEs 108
6.3.1 Coupled 1st Order ODEs 108
6.3.2 Oscillatory Motion 110
6.3.3 More Than One Dimension 116
Exercises 117
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Contents • ix
Chapter 7: Fourier Analysis 119
7.1 The Fourier Series 120
7.2 Fourier Transforms 124
7.3 The Discrete Fourier Transform 127
7.4 The Fast Fourier Transform 129
7.4.1 Brief History and Development 129
7.4.2 Implementation and Sampling 130
Exercises 135
Chapter 8: Monte Carlo Methods 137
8.1 Monte Carlo Integration 137
8.1.1 Dart Throwing 137
8.1.2 General Integration Using
Monte Carlo 143
8.1.3 Importance Sampling 146
8.2 Monte Carlo Simulations 148
8.2.1 Random Walk 148
8.2.2 Radioactive Decay 154
Exercises 156
Chapter 9: Partial Differential Equations 159
9.1 Classes, Boundary Values, and
Initial Conditions 160
9.2 Finite Difference Methods 164
9.2.1 Difference Formulas 165
9.2.2 Application of Difference
Formulas 168
9.3 Richardson Extrapolation 174
9.4 Numerical Methods to Solve PDEs 178
9.4.1 The Heat Equation with Dirichlet
Boundaries 178
9.4.2 The Heat Equation with
Neumann Boundaries 190
9.4.3 The Steady-State Heat Equation 193
9.4.4 The Wave Equation 196
9.5 Pointers To The Finite Element Method 199
Exercises 200
Chapter 10: Advanced Numerical Quadrature 203
10.1 General Quadrature 203
10.2 Orthogonal Polynomials 207
10.3 Gauss–Legendre Quadrature 210
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