{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 2: Non-linear Convection\n",
    "\n",
    "I began this trek along the 12 steps of Navier stokes with the extremely simple 1-D linear convection equation shown below:\n",
    "\n",
    "$$ \\frac{\\partial u}{\\partial t} + c \\frac{\\partial u}{\\partial x} = 0 $$\n",
    "\n",
    "Next, we up the ante and present a new more complex model known as the 1D convection equation (notice the lack of the word linear) :\n",
    "\n",
    "$$ \\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} = 0 $$\n",
    "\n",
    "Now, instead of a constant factor `c` multiplying we have the solution `u` doing that instead. Thus this makes our convection equation behave nonlinearly. We shall follow the exact discretization steps outline in **Step 1** with the following result:\n",
    "\n",
    "$$\\frac{u_i^{n+1} - u^n_i}{\\Delta t} + u^n_i \\frac{u_{i}^{n} - u^n_{i-1}}{\\Delta t} = 0$$\n",
    "\n",
    "Solving for the same unknown value $u_i^{n+1}$:\n",
    "\n",
    "$$ u_i^{n+1} = u^n_i - u^n_i \\frac{\\Delta t}{\\Delta x}(u_i^n - u^n_{i-1}) $$\n",
    "\n",
    "The following step is a condensed version of what we did to set up initial conditions, variables and libraries and is   taken directly from the previous notebook.\n",
    "\n",
    "## Libraries & Initial Conditions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MklVZKETEdcADEzb5deDKiLg7335XVW1ZjaKnUH4oeElOM0tXnWMKTwUOkfR/Jd0o6fXjNpR0lqTtkrbv3r17Jo1bXp+53LcoCwX3FMwsTXWGQgP4eeCVwMuA90p66qgNI+LCiNgaEVs3b948k8YVJ+4yb4gH2RiFewpmlqpGjcfeCdwfEY8Cj0q6DjgB+HaNbeqrsnzU8piCmSWqzp7CXwG/IKkhaX/gecCOGtvzOP2eQunlozmXj8wsWZX1FCRdDpwCHCZpJ3AOsAAQERdExA5J1wC3AD3googYe/nqrDUru/ponvsfeazUfZqZlaWyUIiIM6bY5sPAh6tqw1q0OkX5qOSeQsPlIzNLl2c0j1H0FBZLHmjOykceaDazNDkUxijuZFrmymvgS1LNLG0OhTGqm9HsUDCzdDkUxqhsnsLCnNdTMLNkORTGaLZ7zAkW5lXqfpca8zzW6dHrRan7NTMrg0NhjGLVNankUCgW2nFvwcwS5FAYo+wFdgpeaMfMUuZQGKPZ7rFU4loKhSJofPtsM0uRQ2GMonxUtuWegstHZpYeh8IYzXaPxSpCIb+ayeUjM0uRQ2GMVqdb+i0uYKB85FAwswQ5FMZotrulz1GA5TWfXT4ysxQ5FMZotnuV9BQWPdBsZglzKIxR9UBzy+UjM0uQQ2GM6uYpFGMKLh+ZWXqmWk9B0vtGPR4Rv19uc9JRVfnIA81mlrJpF9l5dOD7JeA0Elo6swrNdrf0tRSA/oQ4h4KZpWiqUIiIjwz+LOmPgS9U0qJEtNq9astHvveRmSVotfWR/YGjymxISrq94LGuy0dmtv5MO6ZwK1Dc63ke2Azss+MJxRrKVfQU5ufEwrw80GxmSZp2TOG0ge87wA8iolNBe5JQnLCruCFetl+vvmZmaZp2TOGuqhuSkmZFS3EWFhfm+70RM7OUeJ7CCMUCOFWFwtLCHC2Xj8wsQQ6FEYqewmJV5aOFed/mwsyS5FAYoery0dLCnAeazSxJDoURihP2YgWXpIIHms0sXQ6FEZoVXpJa7NehYGYpciiMUNzBtIr1FMDlIzNLl0NhhP48hYrKR4seaDazRDkURqh8oLkx70tSzSxJDoURZnP1kXsKZpYeh8IIzU615SMPNJtZqhwKIzRnMdDsW2ebWYIcCiM02z02zM8xN6dK9r/UmKfbC9pdB4OZpcWhMEKz3a1s4hp4TQUzS1dlZz5JF0vaJem2FbZ7jqSOpF+tqi17qtXpVjbIDMtjFZ6rYGapqbKnsA04ddIGkuaBPwK+WGE79lizXc2qa4VF9xTMLFGVnfki4jrggRU2+23gc8CuqtqxGs12t7JBZli++6rXVDCz1NQ2piBpC/ArwPlTbHuWpO2Stu/evbvytjXbVZePip6Cy0dmlpY6B5r/BHhHRKx4ZoyICyNia0Rs3bx5c+UNq7p85IFmM0vVtGs0V2ErcIUkgMOAV0jqRMRf1tgmILtL6qbF6t6aYu1n9xTMLDW1hUJEHFN8L2kb8PkUAgGyk/WhG2dRPnJPwczSUlkoSLocOAU4TNJO4BxgASAiLqjquGVotbuzKR95oNnMElNZKETEGXuw7RuqasdqVD/Q7PKRmaXJM5pHaHVmM9DsS1LNLDUOhRGqnqdQ7Ns9BTNLjUNhhGanV+m9jxb75SP3FMwsLQ6FIe1uj24vKp/RLC2vBW1mlgqHwpCqV10DkMRiw2sqmFl6HApDijp/lQPN2f69+pqZpcehMKQ4US9W2FOAbLDZoWBmqXEoDCkuE62yfJTtf85XH5lZchwKQ/rlo4bLR2a2/jgUhsxioBmy8pQHms0sNQ6FIcsDzVWPKcy5p2BmyXEoDFnuKVRfPvI8BTNLjUNhSNMDzWa2jjkUhiwPNFcdCvO+dbaZJcehMGRm5SPPUzCzBDkUhsxs8prLR2aWIIfCkFbHt7kws/XLoTCk2e4iwYb5at+axYV5Wp0eEVHpcczM9oRDYUixwI6kSo9T9ERansBmZglxKAxptqtdirOw2F99zSUkM0uHQ2FIs92tfI4CLPcUPNhsZilxKAxpdnqzCQX3FMwsQQ6FIc12l8WK75AKyzOmPYHNzFLiUBjSmlVPoRhodvnIzBLiUBiSjSnMsKfg8pGZJcShMKTV7vavDKpSf6DZl6SaWUIcCkN8SaqZrWcOhSHNzqwuSXUomFl6HApDihnNVfNAs5mlyKEwZFblI1+SamYpcigMmd2MZpePzCw9DoUBEUGr06t8LQWApYZvc2Fm6XEoDJjVWgoAjfk5GnNyT8HMkuJQGNBfinMGA81QLLTjnoKZpaOyUJB0saRdkm4b8/xrJd0i6VZJ10s6oaq2TKs4Qc9iTCE7zpwHms0sKVX2FLYBp054/rvAv4yIZwLvBy6ssC1T6fcUZlA+gmwCm8tHZpaSRlU7jojrJB094fnrB378OnBUVW2ZVvGpfZY9Bc9TMLOUpDKm8Ebgr8c9KeksSdslbd+9e3dljVguH83mbcnGFNxTMLN01B4Kkn6JLBTeMW6biLgwIrZGxNbNmzdX1pZaBpo9pmBmCamsfDQNSc8CLgJeHhH319kWWA6FWcxTgHyg2eUjM0tIbT0FSU8CrgReFxHfrqsdg2ZePvJAs5klprKegqTLgVOAwyTtBM4BFgAi4gLgfcChwCckAXQiYmtV7ZlGa+YDzQ4FM0tLlVcfnbHC828C3lTV8Vdj+ZLU2YTCostHZpaY2geaU9IvHzVmd/VRywPNZpYQh8KAmfcUGu4pmFlaHAoDZn+bC48pmFlaHAoDWp0uC/Nifk4zOd5SY55OL+h03VswszQ4FAY0272ZTVyDgSU5Ow4FM0uDQ2FAs9NlcUZzFMCrr5lZehwKA5rtLos19BSa7imYWSIcCgNa7d7MZjODewpmlh6HwoBmuzuzK4+Afq/EoWBmqXAoDGh2ZhsK/fKR5yqYWSIcCgOaNZWPWu4pmFkiHAoDmu3ujC9JzctHvtWFmSXCoTBg1mMKLh+ZWWocCgOa7d5s5yl4oNnMEuNQGNCa+UBzEQruKZhZGhwKA+q6zYV7CmaWCofCgGxMoYbJax5oNrNEOBRynW6PTi9mPHnNA81mlhaHQq64/9AsewqSWGzMeZ6CmSXDoZCb9aprBS+0Y2YpcSjk+qEww4FmyHomLh+ZWSocCrnixDzLeQqQ9xQ80GxmiXAo5GorHzVcPjKzdDgUcq1OXWMKLh+ZWTocCrnixLzUmO1bsuiBZjNLiEMhV+vVR16O08wS4VDItfrzFGY9puB5CmaWDodCbrmnMPvyUcs9BTNLhEMh178kddbzFBpzHlMws2Q4FHJ19RQ8o9nMUuJQyDV9SaqZmUOhsFw+qmdGc0TM9LhmZqM4FHKtdpfFxhySZnrcpYV5IuCxrnsLZlY/h0IuW2BntqUj8JoKZpYWh0Ku2e7NfJAZlscwPFfBzFJQ2VlQ0sWSdkm6bczzkvSnku6UdIukZ1fVlmk0O/X0FPpLcrqnYGYJqPKj8Tbg1AnPvxw4Nv86Czi/wrasqNnuznwtBVi+BNa3zzazFDSq2nFEXCfp6AmbnA58OrLLbr4u6WBJR0TE96tozzW33cfZn7l57PM/bXd55lEHV3HoifbfkAXRq877KvMzHuQ2s73LG3/hGH73pU+r9BiVhcIUtgD3DPy8M3/sZ0JB0llkvQmARyT9wyqPeRjww3FPfhfQW1e553RNfM37KL/m9WHdveaz4bCzV/+anzzNRnWGwtQi4kLgwrXuR9L2iNhaQpP2Gn7N64Nf8/owi9dc59VH3wOeOPDzUfljZmZWkzpD4Srg9flVSCcDD1U1nmBmZtOprHwk6XLgFOAwSTuBc4AFgIi4ALgaeAVwJ/AT4Myq2jJgzSWovZBf8/rg17w+VP6a5XvumJlZwTOazcysz6FgZmZ96yYUJJ0q6R/y22q8s+72VE3SEyV9WdIdkm6X9La62zQLkuYl/b2kz9fdllnJJ35+VtK3JO2Q9Py621QlSf8l/2/6NkmXS1qqu01VGHWrIElPkPQ3kv4x//eQso+7LkJB0jzwcbJbazwdOEPS0+ttVeU6wNkR8XTgZOAt6+A1A7wN2FF3I2bsY8A1EXEccAL78OuXtAX4z8DWiHgGMA+8pt5WVWYbP3uroHcCfxsRxwJ/m/9cqnURCsBzgTsj4jsR8RhwBdltNvZZEfH9iLgp//7HZCeKLfW2qlqSjgJeCVxUd1tmRdJBwL8APgkQEY9FxIP1tqpyDWA/SQ1gf+DemttTiYi4Dnhg6OHTgU/l338K+NdlH3e9hMK4W2qsC/k9qE4Cbqi3JZX7E+DtwHq65ewxwG7gf+Rls4skbay7UVWJiO8BfwzcTXZLnIci4ov1tmqmDh+Yz3UfcHjZB1gvobBuSdoEfA74nYh4uO72VEXSacCuiLix7rbMWAN4NnB+RJwEPEoFJYVU5DX008nC8Ehgo6TfqLdV9chvJlr6nIL1Egrr8pYakhbIAuHSiLiy7vZU7IXAqyT9E1l58EWSLqm3STOxE9gZEUUv8LNkIbGvegnw3YjYHRFt4ErgBTW3aZZ+IOkIgPzfXWUfYL2EwjeAYyUdI2kD2cDUVTW3qVLKFpv+JLAjIj5ad3uqFhHvioijIuJosr/vlyJin/8EGRH3AfdIKu6n/GLgjhqbVLW7gZMl7Z//N/5i9uGB9RGuAn4z//43gb8q+wB7xV1S1yoiOpLeCnyB7GqFiyPi9pqbVbUXAq8DbpVULCTx7oi4usY2WTV+G7g0/8DzHWZzy5haRMQNkj4L3ER2hd3fs4/e7mLMrYI+BHxG0huBu4BXl35c3+bCzMwK66V8ZGZmU3AomJlZn0PBzMz6HApmZtbnUDAzsz6HgpmZ9TkUzMysz6FgtkaSniPpFklLkjbm9/p/Rt3tMlsNT14zK4GkDwBLwH5k9yL6YM1NMlsVh4JZCfJbTHwDaAIviIhuzU0yWxWXj8zKcSiwCTiArMdgtldyT8GsBJKuIrtl9zHAERHx1pqbZLYq6+IuqWZVkvR6oB0Rl+XrgV8v6UUR8aW622a2p9xTMDOzPo8pmJlZn0PBzMz6HApmZtbnUDAzsz6HgpmZ9TkUzMysz6FgZmZ9/x+eQDOV31UdfQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Adding inline command to make plots appear under comments\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import time, sys\n",
    "%matplotlib inline   \n",
    "\n",
    "#Same initial conditions as in step 1\n",
    "grid_length = 10\n",
    "grid_points = 41\n",
    "dx = grid_length / (grid_points - 1) \n",
    "nt = 400\n",
    "dt = 0.025\n",
    "\n",
    "#Initiallizing the shape of the wave to the same one from step 1 and displaying it\n",
    "u = np.ones(grid_points)\n",
    "u[int(.5/ dx):int(1 / dx + 1)] = 2\n",
    "plt.plot(np.linspace(0,grid_length,grid_points), u);\n",
    "plt.ylim(1,2);\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('u')\n",
    "plt.title('1D Non-Linear Convection t=0');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Applying the discretization\n",
    "\n",
    "Now we apply the discretization as outlined above and check out the final results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "un = np.ones(grid_points)\n",
    "\n",
    "for n in range(nt): #Runs however many timesteps you set earlier\n",
    "    un = u.copy()   #copy the u array to not overwrite values\n",
    "    for i in range(1,grid_points):\n",
    "        u[i] = un[i] - un[i] * dt/dx * (un[i]-un[i-1]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.linspace(0,grid_length,grid_points), u);\n",
    "plt.ylim(1,2);\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('u')\n",
    "plt.title('1D Non-Linear Convection t=10');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Results\n",
    "\n",
    "Looks pretty unchanged, to have a better idea if it behaved the same way we will take a look at the animation.\n",
    "\n",
    "## Animating the wave moving"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Imports for animation and display within a jupyter notebook\n",
    "from matplotlib import animation, rc \n",
    "from IPython.display import HTML\n",
    "\n",
    "#Generating the figure that will contain the animation\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlim(( 0, grid_length))\n",
    "ax.set_ylim((1, 2))\n",
    "line, = ax.plot([], [], lw=2)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('u')\n",
    "plt.title('1D Non-Linear Convection time evolution from t=0 to t=10');\n",
    "#Resetting the U wave back to initial conditions\n",
    "u = np.ones(grid_points)\n",
    "u[int(.5/ dx):int(1 / dx + 1)] = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Initialization function for funcanimation\n",
    "def init():\n",
    "    line.set_data([], [])\n",
    "    return (line,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Main animation function, each frame represents a time step in our calculation\n",
    "def animate(j):\n",
    "    x = np.linspace(0, grid_length, grid_points)\n",
    "    un = u.copy()   #copy the u array to not overwrite values\n",
    "    for i in range(1,grid_points):\n",
    "        u[i] = un[i] - un[i] * dt/dx * (un[i]-un[i-1]) \n",
    "    line.set_data(x, u)\n",
    "    return (line,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    document.getElementById(this.slider_id).max = this.frames.length - 1;\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<div class=\"animation\" align=\"center\">\n",
       "    <img id=\"_anim_imgf286a0e4e6e0400c8b0f4ac1496f0cd3\">\n",
       "    <br>\n",
       "    <input id=\"_anim_sliderf286a0e4e6e0400c8b0f4ac1496f0cd3\" type=\"range\" style=\"width:350px\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           onchange=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.set_frame(parseInt(this.value));\"></input>\n",
       "    <br>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.previous_frame()\">\n",
       "        <i class=\"fa fa-step-backward\"></i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.reverse_animation()\">\n",
       "        <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "        </i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "        </button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"animf286a0e4e6e0400c8b0f4ac1496f0cd3.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "  <form action=\"#n\" name=\"_anim_loop_selectf286a0e4e6e0400c8b0f4ac1496f0cd3\" class=\"anim_control\">\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"once\" > Once </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"loop\" checked> Loop </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"reflect\" > Reflect </input>\n",
       "  </form>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_imgf286a0e4e6e0400c8b0f4ac1496f0cd3\";\n",
       "    var slider_id = \"_anim_sliderf286a0e4e6e0400c8b0f4ac1496f0cd3\";\n",
       "    var loop_select_id = \"_anim_loop_selectf286a0e4e6e0400c8b0f4ac1496f0cd3\";\n",
       "    var frames = new Array(0);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
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       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        animf286a0e4e6e0400c8b0f4ac1496f0cd3 = new Animation(frames, img_id, slider_id, 20.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=nt, interval=20)\n",
    "HTML(anim.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "Yep, pretty similar behaviour. Only difference would be in the speed in which it changes shape in the beginning steps, but not much else is of notice.\n",
    "\n",
    " Next we will take a quick detour to talk about the CFL condition."
   ]
  }
 ],
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