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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}