{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Practical Algorithms I - Euler Method\n", "\n", "We've covered algorithms in the abstract sense previously. Now we will look at a concrete, basic one that can be used for physical problems. We will look at the Euler method. It works as follows:\n", "\n", "For the situation \n", "$$ y' = f(t,y(t)) $$ \n", "and \n", "$$ y(t_0) = y_0$$ we will let \n", "$$ y(t_0 + \\Delta t) = y(t_0) + f(t_0, y(t_0)) \\Delta t $$\n", "for some small $ \\Delta t $. We will apply it in the context of acceleration and velocity, where the algorithm will be as follows:\n", "$$ v(t_0) = v_0$$ we will let \n", "$$ v(t_0 + \\Delta t) = v(t_0) + a(t_0, v(t_0)) \\Delta t $$\n", "\n", "We will apply this method to the situation of a rocket, as described in the lectures. To do this, first we will import the neccesary modules:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import copy\n", "from math import log\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will then define functions which step the time forwards, implementing the Euler method, and a method to calculate the analytic solution to the velocity at a certain mass, as defined in lectures:\n", "(Keep in mind global variables are generally bad practice. They are used here because it allows for short code and the functions are highly specific to this rocket problem, meaning it's highly unlikely they'll be reused, making it not so much of a problem)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def step_time():\n", " global m_change_rate, delta_t, m_rocket, v_ejecta, v_rocket\n", " m_change = m_change_rate * delta_t\n", " m_rocket += m_change\n", " force = - m_change_rate * v_ejecta\n", " accel = force/m_rocket\n", " v_rocket += accel*delta_t\n", "\n", "def analytic_soln():\n", " global m_change_rate, delta_t, m_rocket, v_ejecta, v_rocket\n", " return v_ejecta * log(initial_m_rocket / m_rocket)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will then run the simulation for max_t time using a while loop. The results are appended to the arrays t, v, and m for plotting at a later time. The code should be self explanatory as variable names are as described in lectures. Read the code and make sure you understand it." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [], "source": [ "m_change_rate = -0.1\n", "v_ejecta = 5.\n", "delta_t = 5.\n", "m_rocket = 10.\n", "initial_m_rocket = copy.copy(m_rocket)\n", "v_rocket = 0.\n", "t = []\n", "v = []\n", "m = []\n", "analytic = []\n", "current_t = 0.\n", "max_t = 95.\n", "\n", "while current_t < max_t:\n", " step_time()\n", " t.append(current_t)\n", " v.append(v_rocket)\n", " m.append(m_rocket)\n", " analytic.append(analytic_soln())\n", " current_t += delta_t" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We will then plot the results in publication-quality figures. Try playing with delta_t in the cell above and see how it affects how well the Euler method approximates the real function." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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