PHYS248: Computer Assisted Mathematics and Physics

A second-year course Introduction to Computational Physics expressed in the Python programing language.

(c) Copyright 2020 Falk Herwig (University of Victoria)

Index of PHYS248

Topics, week-by-week

1. Python programming I: Language elements, basic math operations
2. Python programming II: Plotting, interpolation, units, examples and applications
3. Linux/Unix OS, GNU, editors, git
4. Numerical integration and differentiation
5. Python programming III: program organization (functions, modules, classes, libraries, scripts, parameter passing)
6. Numerical Analysis: Integration and differentiation with libraries
7. ODE I: Skydiver problem or similar, explicit Euler-step integration, mulit-processing
8. ODE II, Linear transformations and linear algebra
9. Simple stats & data modeling
10. Data analysis and examples
11. Sympy (symbolic computing) and image processing
12. Optimization (simulated annealing), audification, LIGO graviational wave signals

1.1 Intro to Python language I (1/2)

• Intro
  • JupyterLab and notebooks
  • Using a code cell like a calculator
  • Variables
• Algebra and statements
  • Arithmetic operators
  • Difference between mathematical equality and computational statment
  • How to solve an exercise?
• Data types
  • Overview
  • Strings and scalar variables
  • Lists and arrays
• Formatted printing

1.2 Intro to Python language I (2/2)

• Data types II
  • Type of a variable and type conversion
  • Boolean
  • Array review: slicing, index arrays, masks
• Intro to libraries
  • Loading libraries, Python name space and the doc string
  • Four different ways to do sqrt
• Flow control
  • if, for loop, while, try
  • Avoiding loops

2.1 Intro to Python language II (1/2)

• Simple line plots and examples
  • Convergence of geometric series
• Advanced arrays
  • Array analysis
  • A selection of array functions
  • Higher-dimensional arrays

2.2 Intro to Python language II (2/2)

• Advanced plotting
  • Bar plots, scatter plots
  • 2D plots
• Basic I/O
  • Interactive input
  • Writing and reading ASCII tables

3.1 Linux/Unix OS, GNU, editors, git

• JupyterLab
  • Notebooks
  • Terminal, Markdown documents and text editor
  • Restarting the JupyetLab application
• Software and hardware
• Introduction to the command line
  • Shell
  • Basic file system commands
  • Networking and compressing data
  • Command line editor
• Distributed version control - Git
  • Basic concepts
  • The most important git commands

3.2 Linux/Unix OS, GNU, editors, git

• Review
  • Terminal FAQs
  • Git FAQs
• Git - part II
  • Additional useful commands, branching and merging
  • How to get and stay out of trouble
• Advanced terminal and shell commands
  • More file system commands
  • More about the shell
  • Networking
  • Shell scripts
• More git
• The sys module

4.1 Numerical integration and differentiation

• The Riemann sum
  • Midpoint rule
  • Trapezoidal rule

4.2 Numerical integration and differentiation

• Introduction to functions
  • def
  • lambda
• Numerical derivatives
  • Difference equations
  • Derivatives
  • Errors: accuracy vs. precision
  • Convergence
  • Higher-order derivative

5.1 Python programming III: program organization

• Functions an modules
  • Try - except
  • Combine functions into a module for later use
• Python scripts
  • Module as Python script - the test block
  • A module directory
• Units and constants
• Miscellaneous
  • Dictionaries

5.2 Python programming III: program organization

• Review:
  • units
  • dictionaries
  • An example where Monte-Carlo integration wins
• Integration with libraries
• Miscellaneous
  • sort, join, split, strip

6.1 Numerical Analysis: Integration and differentiation with libraries

• Derivatives
  • Derivative of numerical data
  • Non-equidistant and noisy data
  • Gradient of 2D function
• Miscellaneous
  • map
  • Multi-threaded processing
  • Animation, make movie with ffmpeg

6.2 Midterm

7.1 ODE I

• Ordinary differential equations
  • Euler step
  • Discretisation
• Miscellaneous
  • map
  • Multi-threaded processing
  • Animation, make movie with ffmpeg

7.2 ODE I

• Non-linear equations
  • Relaxation method
  • Binary search - bisection method
  • Newton-Raphson
• Skydiver problem: Falling body with drag
  • Equation of motion
  • Solve ODE explicit
  • Solve with library

8.1 ODE II

• 3D line plots
• ODE's with mulitple coupled equations
  • Lotka–Volterra equations and comparison of two solvers
  • Discuss and understand the accuracy of a numerical solution, and how to use libraries properly
  • Conclusions and recommendations
• Chaos: Lorenz equations

8.2 Linear transformations and linear algebra

• Linear Algebra
  • Basic matrix and vector operations
  • Linear transformations
  • Solving systems of equations

9.1 Data modeling

• Fitting data
• Standard normal distribution
• Moments of distribution
• Least-square fitting of arbitrary curve

9.2 Data modeling

• Linear correlation
• Fitting data with a model

10.1 Data analysis

• Fitting a Gaussian distribution to a data set
• Application: Fitting mixing results from hydrodynamic simulations

10.2 Examples

• A couple of miscellaneous items: plotting lines with colormap color, integer word length and sets
• Lorenz equations
• Recaman's sequence

11.1 Sympy

• An introduction to Sympy

11.2 More Sympy and image processing

• More Sympy
  • limits
  • series expansion
  • root finding
  • differential equations
• Image Processing for fun and science!
  • image basics
  • linear image filters
  • other filters (median)
  • histograms and statistics

12.1 Optimization: Simulated Annealing

• Simulated Annealing
  • Travelling salesman problem

12.2 Audification, LIGO Gravitational Wave Signals

• Audification
• The LIGO gravitational wave discovery in 2015