Exam (elaborations)

Richard fitzpatrick computational physics

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Richard Fitzpatrick Computational Physics

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richard fitzpatrick computational physics

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Computational Physics
Richard Fitzpatrick
Professor of Physics
The University of Texas at Austin Contents
1 Introduction 8
1.1 Intended Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Major Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Purpose of Course . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Course Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Programming Methodologies . . . . . . . . . . . . . . . . . . . . . 9
1.6 Scientiﬁc Programming Languages . . . . . . . . . . . . . . . . . . 11
2 Scientiﬁc Programming in C 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Expressions and Statements . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Library Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Data Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Structure of a C Program . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8 Control Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.10 Pointers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.11 Global Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2 2.12 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.13 Character Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.14 Multi-File Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.15 Command Line Parameters . . . . . . . . . . . . . . . . . . . . . . . 77
2.16 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.17 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.18 C++ Extensions to C . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.19 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.20 Variable Size Multi-Dimensional Arrays . . . . . . . . . . . . . . . . 89
2.21 The CAM Graphics Class . . . . . . . . . . . . . . . . . . . . . . . . 93
3 Integration of ODEs 101
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3 Numerical Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4 Numerical Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . 10 6
3.5 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.6 An Example Fixed-Step RK4 routine . . . . . . . . . . . . . . . . . 109
3.7 An Example Calculation . . . . . . . . . . . . . . . . . . . . . . . . 111
3.8 Adaptive Integration Methods . . . . . . . . . . . . . . . . . . . . . 11 3
3.9 An Example Adaptive-Step RK4 Routine . . . . . . . . . . . . . . . 117
3.10 Advanced Integration Methods . . . . . . . . . . . . . . . . . . . . 12 1
3 3.11 The Physics of Baseball Pitching . . . . . . . . . . . . . . . . . . . . 121
3.12 Air Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.13 The Magnus Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.14 Simulations of Baseball Pitches . . . . . . . . . . . . . . . . . . . . 1 27
3.15 The Knuckleball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4 The Chaotic Pendulum 140
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2 Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.4 Validation of Numerical Solutions . . . . . . . . . . . . . . . . . . . 1 48
4.5 The Poincar ´e Section . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.6 Spatial Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . 152
4.7 Basins of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.8 Period-Doubling Bifurcations . . . . . . . . . . . . . . . . . . . . . 16 3
4.9 The Route to Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.10 Sensitivity to Initial Conditions . . . . . . . . . . . . . . . . . . . . 173
4.11 The Deﬁnition of Chaos . . . . . . . . . . . . . . . . . . . . . . . . 179
4.12 Periodic Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.13 Further Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5 Poisson’s Equation 189
4

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