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\documentclass[10pt,a4paper]{report}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{units}
\usepackage{fancyvrb}
\fvset{fontsize=\normalsize}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\usepackage{multicol}
\title{CFD Python Modules}
\begin{document}
\section*{Quick Python Intro}
A \textit{very} quick crash-course in the basics of Python, broadly covering
\begin{itemize}
\item[] Libraries (NumPy and Matplotlib, specifically)
\item[] Variables
\item[] Whitespace
\item[] Array slicing and assigment
\end{itemize}
\section*{Step 1: 1-D Linear Convection}
\begin{equation}
\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0
\end{equation}
\subsection*{Math}
This section introduces the reader to the ``parts`` of the PDE and how to discretize them.
\begin{itemize}
\item[] Introduce the idea of a grid
\item[] Expand equation using the definition of a derivative
\item[] Discretize into small chunks
\item[] Re-arrange to solve for $u^{n+1}_i$
\item[] Initial and boundary conditions
\end{itemize}
\subsection*{Python}
\begin{itemize}
\item[] Importing libraries
\item[] Assigning variables
\item[] Basic 2D plotting
\item[] Simple for-loops
\end{itemize}
\vspace{.5cm}
YouTube videos on order of convergence, truncation error, etc...
\section*{Step 2: 1-D Nonlinear Convection}
\begin{equation}
\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0
\end{equation}
\subsection*{Math}
\begin{itemize}
\item[] Introduce non-linear PDE equation
\item[] Expand equation using definition of derivative
\item[] Discretize
\item[] Solve for $u^{n+1}_i$
\end{itemize}
\section*{CFL Condition}
A short side-trip into the CFL condition, order of convergence and blowing things up.
\subsection*{Math}
\begin{itemize}
\item[] The Courant number
\item[] Explanation of blow-up behavior when wave travels a distance $> dx$ during a time $dt$
\end{itemize}
\subsection*{Python}
\begin{itemize}
\item[] Quick introduction to defining a function to use code repeatedly
\end{itemize}
\section*{Step 3: 1-D Diffusion}
\begin{equation}
\frac{\partial u}{\partial t} = \nu \frac{\partial ^2 u}{\partial x^2}
\end{equation}
\subsection*{Math}
\begin{itemize}
\item[] Introduce diffusion equation
\item[] Discretize 2nd order derivative using Taylor series expansion
\item[] Discretize time derivative using def. of derivative
\end{itemize}
\subsection*{Python}
Nothing new, still no functions being used (yet)
\section*{Step 4: 1-D Burgers' Equation}
\begin{equation}
\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial ^2 u}{\partial x^2}
\end{equation}
\subsection*{Math}
\begin{itemize}
\item[] Introduce Burgers' Equation
\item[] Note that it is combination of diffusion and non-linear convection
\item[] Introduce different I.C. and B.C. for periodic behavior
\begin{itemize}
\item[] e.g. What does $u^{n}_{i+1}$ \textit{mean} at the end of the frame?
\end{itemize}
\end{itemize}
\subsection*{Python}
\begin{itemize}
\item[] Introduce Sympy
\begin{itemize}
\item[] Pretty printing
\item[] Symbolic solving of derivatives
\item[] Usage of Lambdify to make solutions `accessible' to Numpy
\end{itemize}
\item[] Matplotlib
\begin{itemize}
\item[] Plotting multiple lines per plot
\item[] Setting line styles
\item[] Legends
\end{itemize}
\end{itemize}
\section*{Array Operations}
Another brief interlude to introduce handling calculations with array operations instead of iterating over the entire array.
\subsection*{Python}
\begin{itemize}
\item[] Array operations, slicing and copying
\item[] Note about using the \verb|%%timeit| magic function to compare performance
\end{itemize}
\section*{Step 5: 2D Linear Convection}
\subsection*{Math}
\begin{itemize}
\item[] Introduction to 2D grid
\item[] Extension of current discretization rules into $i,j$ flatland
\item[] Discretize 2D equation and solve for unknown
\end{itemize}
\subsection*{Python}
\begin{itemize}
\item[] meshgrid
\item[] Axes3D
\item[] surf and wireframe plots
\item[] Demonstration that nested for-loop results and array operations results are the same
\end{itemize}
\section*{Step 6: 2D Nonlinear Convection}
\subsection*{Math}
\begin{itemize}
\item[] Introduction of coupled PDEs
\item[] Discretization of two equations
\item[] Solving for both $u^{n+1}_{i,j}$ and $v^{n+1}_{i,j}$
\end{itemize}
\section*{Step 7: 2D Diffusion}
\subsection*{Math}
\begin{itemize}
\item[] Introduction to 2D diffusion equation
\item[] Discretization of equation, etc...
\end{itemize}
\subsection*{Python}
Nothing new, although functions are used to display results (probably should switch this over to \texttt{jsanim})
\section*{Step 8: 2D Burgers'}
\end{document}