p06.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p06: Variable coefficient wave equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_format='svg'\n",
    "from matplotlib.collections import PolyCollection\n",
    "from numpy import pi,linspace,sin,cos,exp,round,zeros,arange,real\n",
    "from numpy.fft import fft,ifft\n",
    "from matplotlib.pyplot import figure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "<Figure size 1200x1000 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "# Set up grid and differentiation matrix:\n",
    "N = 128; h = 2*pi/N; x = h*arange(1,N+1);\n",
    "t = 0.0; dt = h/4.0\n",
    "tmax = 8.0; tplot = 0.15;\n",
    "plotgap = int(round(tplot/dt)); dt = tplot/plotgap;\n",
    "nplots = int(round(tmax/tplot))\n",
    "\n",
    "c = 0.2 + sin(x-1.0)**2\n",
    "v = exp(-100.0*(x-1.0)**2); vold = exp(-100.0*(x-0.2*dt-1.0)**2);\n",
    "\n",
    "# wave numbers\n",
    "ik = 1j*zeros(N)\n",
    "ik[0:N//2+1] = 1j*arange(0,N//2+1)\n",
    "ik[N//2+1:]  = 1j*arange(-N//2+1,0,1)\n",
    "\n",
    "# Time-stepping by leap-frog formula\n",
    "data = []; data.append(list(zip(x, v)))\n",
    "tdata = []; tdata.append(0.0)\n",
    "for i in range(1,nplots):\n",
    "    for n in range(plotgap):\n",
    "        t = t + dt\n",
    "        v_hat = fft(v)\n",
    "        w_hat = ik * v_hat; w_hat[N//2] = 0.0\n",
    "        w = real(ifft(w_hat))\n",
    "        vnew = vold - 2.0*dt*c*w\n",
    "        vold = v; v = vnew;\n",
    "    data.append(list(zip(x, v)))\n",
    "    tdata.append(t);\n",
    "\n",
    "fig = figure(figsize=(12,10))\n",
    "ax = fig.add_subplot(111,projection='3d')\n",
    "poly = PolyCollection(data, closed=False, facecolors='white', edgecolors='red')\n",
    "poly.set_alpha(1)\n",
    "ax.add_collection3d(poly, zs=tdata, zdir='y')\n",
    "ax.set_xlabel('x'); ax.set_ylabel('t'); ax.set_zlabel('v')\n",
    "ax.set_xlim3d(0, 2*pi); ax.set_ylim3d(0, 8); ax.set_zlim3d(0, 4)\n",
    "ax.view_init(40,-70)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Make animation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import animation\n",
    "from matplotlib import rc\n",
    "rc('animation', html='jshtml')\n",
    "\n",
    "# First set up the figure, the axis, and the plot element we want to animate\n",
    "fig = plt.figure()\n",
    "ax = plt.axes(xlim=(0, 2*pi), ylim=(-0.1, 2))\n",
    "line, = ax.plot([], [], lw=2)\n",
    "plt.xlabel('x'); plt.ylabel('u')\n",
    "\n",
    "# initialization function: plot the background of each frame\n",
    "def init():\n",
    "    line.set_data([], [])\n",
    "    return line,\n",
    "\n",
    "# animation function.  This is called sequentially\n",
    "def animate(i):\n",
    "    x, v = zip(*data[i])\n",
    "    line.set_data(x, v)\n",
    "    return line,\n",
    "\n",
    "# call the animator.  blit=True means only re-draw the parts that have changed.\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=len(data), interval=50, blit=True)\n",
    "# Save to file\n",
    "try:\n",
    "    anim.save('p06.mp4', fps=20, extra_args=['-vcodec', 'libx264'])\n",
    "except:\n",
    "    print(\"Cannot save mp4 file\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Use this for inline display with controls\n",
    "anim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Use this for inline display of movie\n",
    "#from IPython.display import HTML\n",
    "#HTML(anim.to_html5_video())"
   ]
  }
 ],
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