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Introduction to Computational Physics

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TOC

1 Introduction
1.1 Intended Audience
1.2 Major Sources
1.3 Purpose of Course
1.4 Course Philosophy
1.5 Programming Methodologies
1.6 Scientific Programming Languages

2 Scientific Programming in C
2.1 Introduction
2.2 Variables
2.3 Expressions and Statements
2.4 Operators
2.5 Library Functions
2.6 Data Input and Output
2.7 Structure of a C Program
2.8 Control Statements
2.9 Functions
2.10 Pointers
2.11 Global Variables
2.12 Arrays
2.13 Character Strings
2.14 Multi-File Programs
2.15 Command Line Parameters
2.16 Timing
2.17 Random Numbers
2.18 C++ Extensions to C
2.19 Complex Numbers
2.20 Variable Size Multi-Dimensional Arrays
2.21 The CAM Graphics Class

3 Integration of ODEs
3.1 Introduction
3.2 Euler’s Method
3.3 Numerical Errors
3.4 Numerical Instabilities
3.5 Runge-Kutta Methods
3.6 An Example Fixed-Step RK4 routine
3.7 An Example Calculation
3.8 Adaptive Integration Methods
3.9 An Example Adaptive-Step RK4 Routine
3.10 Advanced Integration Methods
3.11 The Physics of Baseball Pitching
3.12 Air Drag
3.13 The Magnus Force
3.14 Simulations of Baseball Pitches
3.15 The Knuckleball

4 The Chaotic Pendulum
4.1 Introduction
4.2 Analytic Solution
4.3 Numerical Solution
4.4 Validation of Numerical Solutions
4.5 The Poincare Section
4.6 Spatial Symmetry Breaking
4.7 Basins of Attraction
4.8 Period-Doubling Bifurcations
4.9 The Route to Chaos
4.10 Sensitivity to Initial Conditions
4.11 The Definition of Chaos
4.12 Periodic Windows
4.13 Further Investigation

5 Poisson’s Equation
5.1 Introduction
5.2 1-D Problem with Dirichlet Boundary Conditions
5.3 An Example Tridiagonal Matrix Solving Routine
5.4 1-D problem with Mixed Boundary Conditions
5.5 An Example 1-D Poisson Solving Routine
5.6 An Example Solution of Poisson’s Equation in 1-D
5.7 2-D problem with Dirichlet Boundary Conditions
5.8 2-d Problem with Neumann Boundary Conditions
5.9 The Fast Fourier Transform
5.10 An Example 2-D Poisson Solving Routine
5.11 An Example Solution of Poisson’s Equation in 2-D
5.12 Example 2-D Electrostatic Calculation
5.13 3-D Problems

6 The Diffusion Equation
6.1 Introduction
6.2 1-D Problem with Mixed Boundary Conditions
6.3 An Example 1-D Diffusion Equation Solver
6.4 An Example 1-D Solution of the Diffusion Equation
6.5 von Neumann Stability Analysis
6.6 The Crank-Nicholson Scheme
6.7 An Improved 1-D Diffusion Equation Solver
6.8 An Improved 1-D Solution of the Diffusion Equation
6.9 2-D Problem with Dirichlet Boundary Conditions
6.10 2-D Problem with Neumann Boundary Conditions
6.11 An Example 2-D Diffusion Equation Solver
6.12 An Example 2-D Solution of the Diffusion Equation
6.13 3-D Problems

7 The Wave Equation
7.1 Introduction
7.2 The 1-D Advection Equation
7.3 The Lax Scheme
7.4 The Crank-Nicholson Scheme
7.5 Upwind Differencing
7.6 The 1-D Wave Equation
7.7 The 2-D Resonant Cavity

8 Particle-in-Cell Codes
8.1 Introduction
8.2 Normalization Scheme
8.3 Solution of Electron Equations of Motion
8.4 Evaluation of Electron Number Density
8.5 Solution of Poisson’s Equation
8.6 An example 1-D PIC Code
8.7 Results
8.8 Discussion

9 Monte-Carlo Methods
9.1 Introduction
9.2 Random Numbers
9.3 Distribution Functions
9.4 Monte-Carlo Integration
9.5 The Ising Model

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