From 56080de56184e4bf5e62d91805f1eb420aad7eac Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 15 Oct 2023 23:01:03 +0200 Subject: [PATCH 1/3] Implemented pseudocode for forward Euler and RK4 --- latex/main.pdf | Bin 73209 -> 249290 bytes latex/sections/methods.tex | 149 ++++++++++++++++++++++++++++++++++++- 2 files changed, 148 insertions(+), 1 deletion(-) diff --git a/latex/main.pdf b/latex/main.pdf index ed5a52cc460302f0453e07df750681a66ab12161..6e3a09369a9ae07bb88d6b583cfa2badd1227889 100644 GIT binary patch delta 213714 zcmZshV{;{dwr!)1ZQHgxww-irtHTa=Y}>YWY}>YN+j-|yy<7FF?)x@>!1^%8oa?Ov zVmuRxib6?Df`NsR1D0ZTacC8mosF28*umHumY*M%NzTmP!o`x9jfaafH69d{8gQkn z%O3pi*KaGwhBx^+L z%S#Gk{OB^w3~ONaWJilDXRlb80FLDHr+q!1rVg-FV#_KUAD^|L2O5zJ>(r_S9sd19 z9-VKkY2m5zXKDiEFc+*b90YSGsh*rr9KKNj-Rbn^PvrVN-h7!k;l>(68$`F(UGM31 zdK@>O+?g{#rj=HN##2THEp!dg1oDNFKVvfqeCM!;yRmO`5!fh9JwFj?0Hby?%j4*g zgWGFyJ%j38e&0`X4Flbu@fr)+W-S5VyIi6qW^Zp!&dyhvVS}+^PxzWINb@1grAgek 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+Newton's second law of motion is defined as +\begin{equation} + \frac{d^2\vb{r}}{dt^2} = \frac{\vb{F}\left(t, \frac{d\vb{r}}{dt}, \vb{r}\right)}{m}. + \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 redefine Newton's second law of motion as +\begin{equation} + \begin{split} + \frac{d\vb{r}}{dt} &= \vb{v} \\ + \frac{d\vb{v}}{dt} &= \frac{\vb{F}(t, \vb{v}, \vb{r})}{m}. + \end{split} + \label{eq:coupled} +\end{equation} +\subsection{Forward Euler} + +For a single particle, the forward Euler method for a coupled system is +expressed as + +\begin{equation} + \begin{split} + \vb{r}_{i+1} &= \vb{r}_i + h \cdot \frac{d\vb{r}_i}{dt} = \vb{r}_i + h \cdot \vb{v}_i \\ + \vb{v}_{i+1} &= \vb{v}_i + h \cdot \frac{\vb{v}_i}{dt} = \vb{v}_i + h \cdot \frac{\vb{F}\left( t_i, \vb{v}_i, \vb{r}_i \right)}{m}, + \end{split} +\end{equation} +where $\vb{v}_i$ is the current velocity of the particle, $\vb{r}_i$ is the +current position of the particle, $m$ is the mass of the particle, and $h$ +is a predetermined timestep. + +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 $new\_state \leftarrow particles$ \Comment{Create a copy of the particles} + \For{ $i = 1, 2, \ldots , \left| particles \right|$ } + \State $new\_state_i.\vb{v} \leftarrow particles_i.\vb{v} + dt \cdot \frac{\vb{F}}{m}$ + \State $new\_state_i.\vb{r} \leftarrow particles_i.\vb{r} + dt \cdot particles_i.\vb{v}$ + \EndFor + \State $particles \leftarrow new\_state$ + \EndProcedure + \end{algorithmic} + \end{algorithm} +\end{figure} + +\subsection{4th order Runge-Kutta} + +For a single particle, we can express the 4th order Runge-Kutta (RK4) method as +\begin{equation} + \begin{split} + \vb{v}_{i+1} &= \vb{v}_i + \frac{h}{6} \left( \vb{k}_{\vb{v},1} + + 2\vb{k}_{\vb{v},2} + 2\vb{k}_{\vb{v},3} + \vb{k}_{\vb{v},4} + \right) \\ + \vb{r}_{i+1} &= \vb{r}_i + \frac{h}{6} \left( \vb{k}_{\vb{r},1} + + 2\vb{k}_{\vb{r},2} + 2\vb{k}_{\vb{r},3} + \vb{k}_{\vb{r},4} + \right), + \end{split} +\end{equation} +where +\begin{equation} + \begin{split} + \vb{k}_{\vb{v},1} &= \frac{\vb{F}(t_i, \vb{v}_i, \vb{r}_i)}{m} \\ + \vb{k}_{\vb{r},1} &= \vb{v}_i \\ + \vb{k}_{\vb{v},2} &= \frac{\vb{F}(t_i+\frac{h}{2}, \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2}, \vb{r}_i + h \cdot \frac{\vb{k}_{\vb{r},1}}{2})}{m} \\ + \vb{k}_{\vb{r},2} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2} \\ + \vb{k}_{\vb{v},3} &= \frac{\vb{F}(t_i+\frac{h}{2}, \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},2}}{2}, \vb{r}_i + h \frac{\cdot \vb{k}_{\vb{r},2}}{2})}{m} \\ + \vb{k}_{\vb{r},3} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},2}}{2} \\ + \vb{k}_{\vb{v},4} &= \frac{\vb{F}(t_i+h, \vb{v}_i + h \cdot \vb{k}_{\vb{v},3}, \vb{r}_i + h \cdot \vb{k}_{\vb{r},3})}{m} \\ + \vb{k}_{\vb{r},4} &= \vb{v}_i + h \cdot \frac{\vb{k}_{\vb{v},1}}{2}. + \end{split} +\end{equation} + +\newpage + +\begin{figure} + \begin{algorithm}[H] + \caption{RK4 method} + \label{algo:rk4} + \begin{algorithmic} + \Procedure{Evolve RK4}{$dt$} + \State $N \leftarrow \text{Number of particles inside the Penning trap}$ + \State $orig\_p \leftarrow \text{Copy of particles}$ \Comment{Keep the original 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 $\vb{k}_{\vb{r},1,i} \leftarrow \vb{v}$ + \State $\vb{k}_{\vb{v},1,i} \leftarrow \frac{\vb{F}}{m}$ + + \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 $\vb{k}_{\vb{r},2,i} \leftarrow \vb{v}$ + \State $\vb{k}_{\vb{v},2,i} \leftarrow \frac{\vb{F}}{m}$ + + \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 $\vb{k}_{\vb{r},3,i} \leftarrow \vb{v}$ + \State $\vb{k}_{\vb{v},3,i} \leftarrow \frac{\vb{F}}{m}$ + + \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 $\vb{k}_{\vb{r},4,i} \leftarrow \vb{v}$ + \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} \biblio -\end{document} \ No newline at end of file +\end{document} From 03344c3298af2ac652a9697abf66470253d75c3b Mon Sep 17 00:00:00 2001 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236bbd0..38b1079 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -7,3 +7,12 @@ 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", +} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index fb2b32c..80ca419 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -6,18 +6,31 @@ \section{Methods} -Newton's second law of motion is defined as +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} - \frac{d^2\vb{r}}{dt^2} = \frac{\vb{F}\left(t, \frac{d\vb{r}}{dt}, \vb{r}\right)}{m}. + \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}\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 redefine Newton's second law of motion as +We now rewrite Newton's second law of motion as \begin{equation} \begin{split} - \frac{d\vb{r}}{dt} &= \vb{v} \\ - \frac{d\vb{v}}{dt} &= \frac{\vb{F}(t, \vb{v}, \vb{r})}{m}. + \frac{d\vb{r}_i}{dt} &= \vb{v}_i \\ + \frac{d\vb{v}_i}{dt} &= \frac{\vb{F}(t, \vb{v}_i, \vb{r}_i)}{m_i}. \end{split} \label{eq:coupled} \end{equation} @@ -33,9 +46,8 @@ expressed as \vb{v}_{i+1} &= \vb{v}_i + h \cdot \frac{\vb{v}_i}{dt} = \vb{v}_i + h \cdot \frac{\vb{F}\left( t_i, \vb{v}_i, \vb{r}_i \right)}{m}, \end{split} \end{equation} -where $\vb{v}_i$ is the current velocity of the particle, $\vb{r}_i$ is the -current position of the particle, $m$ is the mass of the particle, and $h$ -is a predetermined timestep. +where $i$ is the current time step of the particle $m$ is the mass of the particle, and $h$ +is the time 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 @@ -50,12 +62,12 @@ Algorithm~\ref{algo:forwardeuler} provides an overview on how that can be achiev \label{algo:forwardeuler} \begin{algorithmic} \Procedure{Evolve forward Euler}{$particles, dt$} - \State $new\_state \leftarrow particles$ \Comment{Create a copy of the particles} - \For{ $i = 1, 2, \ldots , \left| particles \right|$ } - \State $new\_state_i.\vb{v} \leftarrow particles_i.\vb{v} + dt \cdot \frac{\vb{F}}{m}$ - \State $new\_state_i.\vb{r} \leftarrow particles_i.\vb{r} + dt \cdot particles_i.\vb{v}$ + \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 - \State $particles \leftarrow new\_state$ \EndProcedure \end{algorithmic} \end{algorithm} @@ -88,6 +100,19 @@ where \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. +around 8 separate loops to calculate all $\vb{k}$ values + \newpage \begin{figure} From f29424f96ea80947719c3527f98f910b6c89b9e2 Mon Sep 17 00:00:00 2001 From: Cory Date: Mon, 16 Oct 2023 20:58:50 +0200 Subject: [PATCH 3/3] Fix some small things --- latex/main.pdf | Bin 261496 -> 270632 bytes latex/sections/methods.tex | 134 +++++++++++++++++++++++++------------ 2 files changed, 92 insertions(+), 42 deletions(-) diff --git a/latex/main.pdf b/latex/main.pdf index 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