Fix some small things

Janitaws/latex setup

.gitignore

@@ -35,9 +35,10 @@
 *.aux
 *.log
 *.out
-*.bib
 *.synctex.gz
 *.bbl 
+latex/mainNotes.bib
+
 
 # C++ specifics
 src/*

latex/main.tex

@@ -0,0 +1,77 @@
+\documentclass[english,notitlepage,reprint,nofootinbib]{revtex4-1}  % defines the basic parameters of the document
+\usepackage{subfiles}
+% For preview: skriv i terminal: latexmk -pdf -pvc filnavn
+% If you want a single-column, remove "reprint"
+
+% Silence warning of revtex4-1
+\usepackage{silence}
+\WarningFilter{revtex4-1}{Repair the float}
+
+% Allows special characters (including æøå)
+\usepackage[utf8]{inputenc}
+\usepackage{fontawesome}
+% \usepackage[english]{babel}
+
+%% Note that you may need to download some of these packages manually, it depends on your setup.
+%% I recommend downloading TeXMaker, because it includes a large library of the most common packages.
+
+\usepackage{physics,amssymb}  % mathematical symbols (physics imports amsmath)
+
+\usepackage{graphicx}         % include graphics such as plots
+\def\resources{\graphicspath{{"./images/"}}}
+\usepackage{xcolor}           % set colors
+\usepackage{hyperref}         % automagic cross-referencing
+\usepackage{listings}         % display code
+\usepackage{subfigure}        % imports a lot of cool and useful figure commands
+% \usepackage{float}
+%\usepackage[section]{placeins}
+\usepackage{algorithm}
+\usepackage[noend]{algpseudocode}
+\usepackage{subfigure}
+\usepackage{tikz}
+\usetikzlibrary{quantikz}
+% defines the color of hyperref objects
+% Blending two colors:  blue!80!black  =  80% blue and 20% black
+\hypersetup{ % this is just my personal choice, feel free to change things
+    colorlinks,
+    linkcolor={red!50!black},
+    citecolor={blue!50!black},
+    urlcolor={blue!80!black}}
+
+% Biblio stuff
+\def\biblio{\bibliographystyle{plain}\bibliography{./references}}
+
+
+\begin{document}
+
+\title{Project 3}  % self-explanatory
+\author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://github.uio.no/FYS3150-G2-2023/Project-3}} % self-explanatory
+\date{\today}                             % self-explanatory
+\noaffiliation                            % ignore this, but keep it.
+
+% Abstract
+\subfile{./sections/abstract.tex}
+
+
+\maketitle
+
+
+% Introduction
+\subfile{./sections/introduction.tex}
+
+% Methods
+\subfile{./sections/methods.tex}
+
+% Results
+\subfile{./sections/results.tex}
+
+% Conclusion
+\subfile{./sections/conclusion.tex}
+
+\onecolumngrid
+
+\bibliographystyle{plain}
+\bibliography{.references/references}
+
+
+\end{document}

latex/references/references.bib

@@ -0,0 +1,18 @@
+
+@inbook{midpoint_rule,
+	author	= "A.C Faul",
+	title	= "A Concise Introduction to Numerical Analysis",
+	chapter	= "5",
+	pages	= "131",
+	publisher= "CRC Press",
+	year	= "2016",
+}
+
+@article{rk4_method,
+    author = "Morten Hjorth-Jensen",
+    title = "Computational Physics, Lecture Notes Fall 2015",
+    journal = "Department of Physics, University of Oslo",
+    year = "2015",
+    chapter = "8.4",
+    pages = "250--252",
+}

latex/sections/abstract.tex

@@ -0,0 +1,12 @@
+\documentclass[../main.tex]{subfiles}
+\resources % search for graphics in ../res/imgs/
+
+
+\begin{document}
+
+\begin{abstract}
+    Add an abstract for the project?  
+\end{abstract}
+
+\biblio
+\end{document}

latex/sections/conclusion.tex

@@ -0,0 +1,13 @@
+\documentclass[../main.tex]{subfiles}
+\resources % search for graphics in ../res/imgs/
+
+
+\begin{document}
+
+\section{Conclusion}
+
+
+
+
+\biblio
+\end{document}

latex/sections/introduction.tex

@@ -0,0 +1,13 @@
+\documentclass[../main.tex]{subfiles}
+\resources % search for graphics in ../res/imgs/
+
+
+\begin{document}
+
+\section{Introduction}
+
+
+
+
+\biblio
+\end{document}

latex/sections/methods.tex

@@ -0,0 +1,235 @@
+\documentclass[../main.tex]{subfiles}
+\resources % search for graphics in ../res/imgs/
+
+
+\begin{document}
+
+\section{Methods}
+
+\subsection{Units and constants}
+
+Before continuing, we need to define the units we'll be working with.
+Since we are working with particles, we need small units to work with so the
+numbers we are working with aren't so small that they could potentially lead
+to large round-off errors in our simulation. The units that we will use are listed in Table~\ref{tab:units}.
+\begin{table}[H]
+    \begin{center}
+        \begin{tabular}[c]{lll}
+            Dimension & Unit & Symbol \\
+            \hline
+            Length & micrometer & $\mu m$ \\
+            Time & microseconds & $\mu s$ \\
+            Mass & atomic mass unit & $u$ \\
+            Charge & the elementary charge & $e$ \\
+            \hline
+        \end{tabular}
+    \end{center}
+    \caption{The set of units we'll be working with.}\label{tab:units}
+\end{table}
+With these base units, we get
+\begin{equation}
+    k_e = 1.3893533 \cdot 10^5 \frac{u(\mu m)^3}{(\mu s)^2 e},
+\end{equation}
+and we get that the unit for magnetic field strength (Tesla, $T$) and electric potential (Volt, $V$) are
+\begin{equation}
+    \begin{split}
+        T &= 9.64852558 \cdot 10^1 \frac{u}{(\mu s) e} \\
+        V &= 9.64852558 \cdot 10^7 \frac{u (\mu m)^2}{(\mu s)^2 e}. \\
+    \end{split}
+    \label{eq:}
+\end{equation}
+
+\subsection{Dealing with a multi--particle system}
+
+For a multi-particles system, we need to modify $\vb{F}$ to account for the 
+force of other particles in the system acting upon each other. To do that, we
+add another term to $\vb{F}$
+\begin{equation}
+    \vb{F}_i(t, \vb{v}_i, \vb{r}_i) = q_i \vb{E}(t, \vb{r}_i) + q_i \vb{v}_i \cross \vb{B} - \vb{E}_p(t, \vb{r}_i),
+\end{equation}
+where $i$ and $j$ are particle indices and
+\begin{equation}
+    \vb{E}_p(t, \vb{r}_i) = q_i k_e \sum_{j \neq i} 
+    q_j \frac{\vb{r_i} - \vb{r_j}}{\left| \vb{r_i} - \vb{r_j} \right|^3}.
+    \label{eq:}
+\end{equation}
+
+Newton's second law for a particle $i$ is then
+\begin{equation}
+    \frac{d^2\vb{r}_i}{dt^2} = \frac{\vb{F}_i\left(t, \frac{d\vb{r}_i}{dt}, \vb{r_i}\right)}{m_i},
+    \label{eq:newtonlaw2}
+\end{equation}
+We can then rewrite the second order ODE from equation~\ref{eq:newtonlaw2} 
+into a set of coupled first order ODEs. 
+We now rewrite Newton's second law of motion as
+\begin{equation}
+    \begin{split}
+        \frac{d\vb{r}_i}{dt} &= \vb{v}_i \\
+        \frac{d\vb{v}_i}{dt} &= \frac{\vb{F}_i(t, \vb{v}_i, \vb{r}_i)}{m_i}.
+    \end{split}
+    \label{eq:coupled}
+\end{equation}
+
+\subsection{Forward Euler}
+
+For a particle $i$, the forward Euler method for a coupled system is 
+expressed as
+\begin{equation}
+    \begin{split}
+        \vb{r}_{i,j+1} &= \vb{r}_{i,j} + h \cdot \frac{d\vb{r}_{i,j}}{dt} = \vb{r}_{i,j} + h \cdot \vb{v}_{i,j} \\
+        \vb{v}_{i,j+1} &= \vb{v}_{i,j} + h \cdot \frac{\vb{v}_{i,j}}{dt} = \vb{v}_{i,j} + h \cdot \frac{\vb{F}\left( t_{j}, \vb{v}_{i,j}, \vb{r}_{i,j} \right)}{m},
+    \end{split}
+\end{equation}
+for particle $i$ where $j$ is the current time step of the particle, 
+$m$ is the mass of the particle, and $h$ is the step length.
+
+When dealing with a multi-particle system, we need to ensure that we do not
+update the position of any particles until every particle has calculated their
+next step. An easy way of doing this is to create a copy of all the particles,
+then update the copy, and when all the particles have calculated their next 
+step, simply replace the particles with the copies. 
+Algorithm~\ref{algo:forwardeuler} provides an overview on how that can be achieved. % Make this better
+
+\begin{figure}[H]
+    \begin{algorithm}[H]
+    \caption{Forward Euler method}
+    \label{algo:forwardeuler}
+        \begin{algorithmic}
+            \Procedure{Evolve forward Euler}{$particles, dt$}
+            \State $N \leftarrow \text{Number of particles in } particles$
+            \State $a \leftarrow \text{Calculate } \frac{\vb{F_i}}{m_i} \text{ for each particle in } particles$
+            \For{ $i = 1, 2, \ldots , N$ }
+            \State $particles_i.\vb{r} \leftarrow particles_i.\vb{r} + dt \cdot particles_i.\vb{v}$
+            \State $particles_i.\vb{v} \leftarrow particles_i.\vb{v} + dt \cdot a_i$
+            \EndFor
+            \EndProcedure
+        \end{algorithmic}
+    \end{algorithm}
+\end{figure}
+
+\subsection{4th order Runge-Kutta}
+
+For a particle $i$, we can express the 4th order Runge-Kutta (RK4) method as
+\begin{equation}
+    \begin{split}
+        \vb{v}_{i,j+1} &= \vb{v}_{i,j} + \frac{h}{6} \left( \vb{k}_{\vb{v},1,i} 
+            + 2\vb{k}_{\vb{v},2,i} + 2\vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} 
+            \right) \\
+        \vb{r}_{i,j+1} &= \vb{r}_{i,j} + \frac{h}{6} \left( \vb{k}_{\vb{r},1,i} 
+            + 2\vb{k}_{\vb{r},2,i} + 2\vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} 
+            \right),
+    \end{split}
+\end{equation}
+where
+\begin{equation}
+    \begin{split}
+        \vb{k}_{\vb{v},1,i} &= \frac{\vb{F}_i(t_j, \vb{v}_{i,j}, 
+            \vb{r}_{i,j})}{m} \\
+        \vb{k}_{\vb{r},1,i} &= \vb{v}_{i,j} \\
+        \vb{k}_{\vb{v},2,i} &= \frac{\vb{F}_i(t_j+\frac{h}{2}, \vb{v}_{i,j} 
+            + h \cdot \frac{\vb{k}_{\vb{v},1,i}}{2}, \vb{r}_{i,j} 
+            + h \cdot \frac{\vb{k}_{\vb{r},1,i}}{2})}{m} \\
+        \vb{k}_{\vb{r},2,i} &= \vb{v}_{i,j} 
+            + h \cdot \frac{\vb{k}_{\vb{v},1,i}}{2} \\
+        \vb{k}_{\vb{v},3,i} &= \frac{\vb{F}_i(t_j+\frac{h}{2}, \vb{v}_{i,j} 
+            + h \cdot \frac{\vb{k}_{\vb{v},2,i}}{2}, \vb{r}_{i,j} 
+            + h \frac{\cdot \vb{k}_{\vb{r},2,i}}{2})}{m} \\
+        \vb{k}_{\vb{r},3,i} &= \vb{v}_{i,j} 
+            + h \cdot \frac{\vb{k}_{\vb{v},2,i}}{2} \\
+        \vb{k}_{\vb{v},4,i} &= \frac{\vb{F}_i(t_j+h, \vb{v}_{i,j} 
+            + h \cdot \vb{k}_{\vb{v},3,i}, \vb{r}_{i,j} 
+            + h \cdot \vb{k}_{\vb{r},3,i})}{m} \\
+        \vb{k}_{\vb{r},4,i} &= \vb{v}_{i,j} 
+            + h \cdot \frac{\vb{k}_{\vb{v},1,i}}{2}.
+    \end{split}
+\end{equation}
+
+In order to find each $\vb{k}_{\vb{r},i}$ and $\vb{k}_{\vb{v},i}$, 
+we need to first compute all $\vb{k}_{\vb{r},i}$ and $\vb{k}_{\vb{v},i}$ 
+for all particles, then we can update the particles in order to compute 
+$\vb{k}_{\vb{r},i+1}$ and $\vb{k}_{\vb{v},i+1}$. In order for the algorithm
+to work, we need to save a copy of each particle before starting so that we
+can update the particles correctly for each step.
+
+This approach would require 8 loops to be able to complete the calculation since
+we cannot update the particles until after all $\vb{k}$ values have been 
+computed, however if we create a temporary array that holds particles, we can
+put the updated particles in there, and then use that array in the next loop,
+and would reduce the required amount of loops down to 4.
+
+\begin{figure}
+    \begin{algorithm}[H]
+    \caption{RK4 method}
+    \label{algo:rk4}
+        \begin{algorithmic}
+            \Procedure{Evolve RK4}{$particles, dt$}
+            \State $N \leftarrow \text{Number of particles inside the Penning trap}$
+            \State $orig\_p \leftarrow \text{Copy of particles}$ 
+            \State $tmp\_p \leftarrow \text{Array of particles of size }N$
+            \State $\vb{k}_{\vb{r}} \leftarrow \text{2D array of vectors of size } 4 \cross N$
+            \State $\vb{k}_{\vb{v}} \leftarrow \text{2D array of vectors of size } 4 \cross N$
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},1,i} \leftarrow particles_i.\vb{v}$
+            \State $\vb{k}_{\vb{v},1,i} \leftarrow \frac{\vb{F}_i}{m_i}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} 
+                + \frac{dt}{2} \cdot \vb{k}_{\vb{r},1,i}$
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} 
+                + \frac{dt}{2} \cdot \vb{k}_{\vb{v},1,i}$
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Update particles}
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},2,i} \leftarrow particles_i.\vb{v}$
+            \State $\vb{k}_{\vb{v},2,i} \leftarrow \frac{\vb{F}_i}{m_i}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} 
+                + \frac{dt}{2} \cdot \vb{k}_{\vb{r},2,i}$
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} 
+                + \frac{dt}{2} \cdot \vb{k}_{\vb{v},2,i}$
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Update particles}
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},3,i} \leftarrow particles_i.\vb{v}$
+            \State $\vb{k}_{\vb{v},3,i} \leftarrow \frac{\vb{F}_i}{m}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + dt \cdot \vb{k}_{\vb{r},3,i}$
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + dt \cdot \vb{k}_{\vb{v},3,i}$
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Update particles}
+
+            \For{ $i = 1, 2, \ldots, N$ }
+            \State $\vb{k}_{\vb{r},4,i} \leftarrow particles_i.\vb{v}$
+            \State $\vb{k}_{\vb{v},4,i} \leftarrow \frac{\vb{F}}{m}$
+
+            \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{6} 
+                \cdot \left( \vb{k}_{\vb{r},1,i} + \vb{k}_{\vb{r},2,i} 
+                + \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$
+
+            \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{6} 
+                \cdot \left( \vb{k}_{\vb{v},1,i} + \vb{k}_{\vb{v},2,i} 
+                + \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$
+            
+            \EndFor
+
+            \State $particles \leftarrow tmp\_p$ \Comment{Final update}
+            \EndProcedure
+        \end{algorithmic}
+    \end{algorithm}
+    $\vb{F}$ in the algorithm does not take any arguments as it uses the 
+    velocities and positions of the particles inside the array $particles$ to
+    calculate the total force acting on particle $i$.
+\end{figure}
+
+
+\subsection{Testing the simulation}
+
+\subsection{Relative error and error convergance rate}
+
+\biblio
+\end{document}

latex/sections/results.tex

@@ -0,0 +1,13 @@
+\documentclass[../main.tex]{subfiles}
+\resources % search for graphics in ../res/imgs/
+
+
+\begin{document}
+
+\section{Results}
+
+
+
+
+\biblio
+\end{document}