CH_EN_6355_Utah_CFD_Saad/The k-Scheme.ipynb

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    "# Advection using the k-Scheme\n",
    "## CH EN 6355 - Computational Fluid Dynamics\n",
    "**Prof. Tony Saad (<a>www.tsaad.net</a>) <br/>\n",
    "slides at: <a>www.tsaad.net</a><br/>\n",
    "Department of Chemical Engineering <br/>\n",
    "University of Utah**\n",
    "<hr/>"
   ]
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    "Here, we will implement the k-scheme or kappa-schemes for advection. It is easiest to implement this scheme since for different values of k, we recover all sorts of high-order flux approximations. We will assume a positive advecting velocity for illustration purposes.\n",
    "\n",
    "We are solving the constant speed advection equation given by\n",
    "\\begin{equation}\n",
    "u_t = - c u_x = - F_x;\\quad F = cu\n",
    "\\end{equation}\n",
    "We will use a simple Forward Euler explicit method. Using a finite volume integration, we get\n",
    "\\begin{equation}\n",
    "u_i^{n+1} = u_i^n - \\frac{\\Delta t}{\\Delta x} (F_{i+\\tfrac{1}{2}}^n - F_{i-\\tfrac{1}{2}}^n)\n",
    "\\end{equation}\n",
    "For constant grid spacing, the k-Scheme is given by\n",
    "\\begin{equation}\n",
    "{\\phi _f} = {\\phi _{\\rm{C}}} + \\frac{{1 - k}}{4}({\\phi _{\\rm{C}}} - {\\phi _{\\rm{U}}}) + \\frac{{1 + k}}{4}({\\phi _{\\rm{D}}} - {\\phi _{\\rm{C}}})\n",
    "\\end{equation}\n",
    "which, for a positive advecting velocity, gives us\n",
    "\\begin{equation}\n",
    "F_{i + {\\textstyle{1 \\over 2}}}^n = c\\phi _{i + {\\textstyle{1 \\over 2}}}^n = c{\\phi _i} + c\\frac{{1 - k}}{4}({\\phi _i} - {\\phi _{i - 1}}) + c\\frac{{1 + k}}{4}({\\phi _{i + 1}} - {\\phi _i})\n",
    "\\end{equation}"
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    "import numpy as np\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "plt.rcParams['animation.html'] = 'html5'\n",
    "from matplotlib import cm"
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    "def step(x,x0):\n",
    "    x0 = 0.6\n",
    "    x1 = 0.8\n",
    "    result = x - x0\n",
    "    result[x-x1<x1] = 1.0            \n",
    "    result[x<x0] = 0.0\n",
    "    result[x>x1] = 0.0  \n",
    "    return result\n",
    "\n",
    "def gaussian(x,x0):\n",
    "    s = 0.08\n",
    "    s = s*s\n",
    "    result = np.exp( -(x-x0)**2/s)\n",
    "    return result"
   ]
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    "L = 1.0\n",
    "n = 128 # cells\n",
    "dx = L/n # n intervals\n",
    "x = np.linspace(-3*dx/2, L + 3*dx/2, n+4) # include ghost cells - we will include 2 ghost cells on each side for high order schemes\n",
    "\n",
    "# create arrays\n",
    "phi = np.zeros(n+4) # cell centered quantity\n",
    "f = np.zeros(n+4+1) # flux\n",
    "u = np.ones(n+4+1) # velocity field - assumed to live on faces same as flux\n",
    "\n",
    "x0 = 0.3\n",
    "# u0 = np.zeros(N + 2)\n",
    "# u0[1:-1] = np.sin(2*np.pi*x)\n",
    "# u0 = np.zeros(N)\n",
    "# phi0 = np.sin(np.pi*x)\n",
    "phi0 = gaussian(x,x0) + step(x,x0)\n",
    "# u0 = triangle(x,0.5,0.75,1)\n",
    "# u0[0:N//2] = 1.0\n",
    "plt.plot(x,phi0)"
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  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "dt= 0.00390625\n",
      "dx= 0.0078125\n"
     ]
    }
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   "source": [
    "cfl =0.5\n",
    "c = 1.0 # use a negative value for left traveling waves\n",
    "dt = cfl*dx/abs(c)\n",
    "print('dt=',dt)\n",
    "print('dx=',dx)"
   ]
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   "execution_count": 17,
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    "# the k scheme\n",
    "k = 0.5\n",
    "# finite volume implementation with arrays for fluxes\n",
    "t = 0\n",
    "tend= L/abs(c)\n",
    "\n",
    "sol = []\n",
    "sol.append(phi0)\n",
    "ims = []\n",
    "\n",
    "fig = plt.figure(figsize=[5,3],dpi=200)\n",
    "plt.rcParams[\"font.family\"] = \"serif\"\n",
    "plt.rcParams[\"font.size\"] = 10\n",
    "plt.rc('text', usetex=True)\n",
    "\n",
    "# plt.grid()\n",
    "plt.xlim([0.,L])\n",
    "plt.ylim([-0.25,1.25])\n",
    "plt.xlabel('$x$')\n",
    "plt.ylabel('$\\phi$')\n",
    "plt.tight_layout()\n",
    "# plot initial condition\n",
    "plt.plot(x,phi0,'darkred',animated=True)\n",
    "\n",
    "i = 0\n",
    "while t < tend:    \n",
    "    phin = sol[-1]\n",
    "\n",
    "#     if (i%16==0):\n",
    "#         shift = int(np.ceil(c*(t-dt)/dx))\n",
    "#         im = plt.plot(x[2:-2], np.roll(phin[2:-2], -shift) ,'k-o',markevery=2,markersize=3.5,markerfacecolor='deepskyblue',\n",
    "#              markeredgewidth=0.25, markeredgecolor='k',linewidth=0.45, animated=True)\n",
    "#         ims.append(im)\n",
    "    \n",
    "    # impose periodic conditions\n",
    "    phin[-2] = phin[2]\n",
    "    phin[-1] = phin[3]    \n",
    "    phin[0] = phin[-4]        \n",
    "    phin[1] = phin[-3]            \n",
    "    \n",
    "    phi = np.zeros_like(phi0)\n",
    "    \n",
    "    # predictor - take half a step and use upwind\n",
    "    # du/dt = -c*du/dx\n",
    "    if c >= 0:\n",
    "        ϕc = phin[1:-2] # phi upwind\n",
    "    else:\n",
    "        ϕc = phin[2:-1] # phi upwind\n",
    "    \n",
    "    f[2:-2] = c*ϕc\n",
    "    phi[2:-2] = phin[2:-2] - dt/2.0/dx*(f[3:-2] - f[2:-3])\n",
    "    phi[-2] = phi[2]\n",
    "    phi[-1] = phi[3]    \n",
    "    phi[0] = phi[-4]        \n",
    "    phi[1] = phi[-3]                \n",
    "    \n",
    "    # du/dt = -c*du/dx\n",
    "    if c >= 0:\n",
    "        ϕc = phi[1:-2] # phi upwind\n",
    "        ϕu = phi[:-3]  # phi far upwind\n",
    "        ϕd = phi[2:-1] # phi downwind\n",
    "    else:\n",
    "        ϕc = phi[2:-1] # phi upwind\n",
    "        ϕu = phi[3:]  # phi far upwind\n",
    "        ϕd = phi[1:-2] # phi downwind\n",
    "        \n",
    "    f[2:-2] = ϕc + (1-k)/4.0*(ϕc - ϕu) + (1+k)/4.0*(ϕd - ϕc)\n",
    "    f = c*f # multiply the flux by the velocity\n",
    "    # advect\n",
    "    phi[2:-2] = phin[2:-2] - c * dt/dx*(f[3:-2] - f[2:-3]) #+ dt/dx/dx*diffusion\n",
    "    t += dt    \n",
    "    i+=1\n",
    "    sol.append(phi)\n",
    "\n",
    "\n",
    "# plt.annotate('k = '+ str(k), xy=(0.5, 0.8), xytext=(0.015, 0.9),fontsize=8)\n",
    "# plt.legend(('exact','numerical'),loc='upper left',fontsize=7)\n",
    "# ani = animation.ArtistAnimation(fig, ims, interval=100, blit=True,\n",
    "#                                 repeat_delay=1000)\n",
    "\n",
    "# ani.save('k-scheme-'+str(k)+'.mp4',dpi=300,fps=24)"
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  },
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   "cell_type": "code",
   "execution_count": 24,
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    "plt.plot(sol[0], label='initial condition')\n",
    "plt.plot(sol[-1], label='one residence time')\n",
    "plt.legend()\n",
    "plt.grid()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Create Animation in Moving Reference Frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "done!\n"
     ]
    },
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      "text/plain": [
       "<Figure size 600x450 with 1 Axes>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Create Animation in Moving Reference Frame\n",
    "\"\"\"\n",
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "matplotlib.use(\"Agg\")\n",
    "fig, ax = plt.subplots(figsize=(4,3),dpi=150) \n",
    "ax.grid(True)\n",
    "f0 = sol[0]\n",
    "line0, = ax.plot(x[2:-2], f0[2:-2] ,'r-',linewidth=0.75, animated=True)\n",
    "line1, = ax.plot(x[2:-2], f0[2:-2] ,'k-o',markevery=2,markersize=3.5,markerfacecolor='deepskyblue',\n",
    "             markeredgewidth=0.25, markeredgecolor='k',linewidth=0.45, animated=True)\n",
    "\n",
    "ann = ax.annotate('time ='+str(round(t,3))+' s', xy=(2, 1), xytext=(40, 200),xycoords='figure points')\n",
    "ax.annotate('k ='+str(k) + ' (k-scheme)', xy=(2, 1), xytext=(40, 190),xycoords='figure points')\n",
    "plt.tight_layout()\n",
    "\n",
    "\n",
    "def animate_moving(i):\n",
    "    print('time=',i*dt)\n",
    "    t = i*dt\n",
    "    xt = x + i*1.1*c*dt\n",
    "    line0.set_xdata(xt[2:-2])\n",
    "    line1.set_xdata(xt[2:-2])    \n",
    "    ax.axes.set_xlim(xt[0],0.0*dx + xt[-1])\n",
    "    f = sol[i]\n",
    "    ax.axes.set_ylim(1.1*min(f) - 0.1,1.1*max(f))\n",
    "    ann.set_text('time ='+str(round(t,4))+'s (' + str(i)+ ').')\n",
    "    shift =int(np.ceil(i*c*dt/dx))\n",
    "    line1.set_ydata(np.roll(f[2:-2], -shift))\n",
    "\n",
    "    f0 = sol[0]\n",
    "    line0.set_ydata(f0[2:-2])\n",
    "    return line0,line1\n",
    "\n",
    "\n",
    "# Init only required for blitting to give a clean slate.\n",
    "def init():\n",
    "    line0.set_ydata(np.ma.array(x[2:-2], mask=True))\n",
    "    line1.set_ydata(np.ma.array(x[2:-2], mask=True))    \n",
    "    return line0,line1\n",
    "\n",
    "ani = animation.FuncAnimation(fig, animate_moving, np.arange(0,len(sol),2*int(1/cfl)), init_func=init,\n",
    "                              interval=20, blit=False)\n",
    "print('done!')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time= 0.0\n",
      "time= 0.0078125\n",
      "time= 0.015625\n",
      "time= 0.0234375\n",
      "time= 0.03125\n",
      "time= 0.0390625\n",
      "time= 0.046875\n",
      "time= 0.0546875\n",
      "time= 0.0625\n",
      "time= 0.0703125\n",
      "time= 0.078125\n",
      "time= 0.0859375\n",
      "time= 0.09375\n",
      "time= 0.1015625\n",
      "time= 0.109375\n",
      "time= 0.1171875\n",
      "time= 0.125\n",
      "time= 0.1328125\n",
      "time= 0.140625\n",
      "time= 0.1484375\n",
      "time= 0.15625\n",
      "time= 0.1640625\n",
      "time= 0.171875\n"
     ]
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-29-f14b34654064>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mani\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msave\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'__k-scheme_'\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;34m'.mp4'\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mfps\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m24\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mdpi\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m300\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/animation.py\u001b[0m in \u001b[0;36msave\u001b[0;34m(self, filename, writer, fps, dpi, codec, bitrate, extra_args, metadata, extra_anim, savefig_kwargs)\u001b[0m\n\u001b[1;32m   1172\u001b[0m                         \u001b[0;31m# TODO: See if turning off blit is really necessary\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1173\u001b[0m                         \u001b[0manim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_draw_next_frame\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mblit\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1174\u001b[0;31m                     \u001b[0mwriter\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgrab_frame\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0msavefig_kwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1175\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1176\u001b[0m         \u001b[0;31m# Reconnect signal for first draw if necessary\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/animation.py\u001b[0m in \u001b[0;36mgrab_frame\u001b[0;34m(self, **savefig_kwargs)\u001b[0m\n\u001b[1;32m    374\u001b[0m             \u001b[0;31m# frame format and dpi.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    375\u001b[0m             self.fig.savefig(self._frame_sink(), format=self.frame_format,\n\u001b[0;32m--> 376\u001b[0;31m                              dpi=self.dpi, **savefig_kwargs)\n\u001b[0m\u001b[1;32m    377\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mRuntimeError\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mIOError\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    378\u001b[0m             \u001b[0mout\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0merr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_proc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcommunicate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/figure.py\u001b[0m in \u001b[0;36msavefig\u001b[0;34m(self, fname, frameon, transparent, **kwargs)\u001b[0m\n\u001b[1;32m   2092\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_frameon\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mframeon\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2093\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2094\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcanvas\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mprint_figure\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfname\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2095\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2096\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mframeon\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/backend_bases.py\u001b[0m in \u001b[0;36mprint_figure\u001b[0;34m(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, **kwargs)\u001b[0m\n\u001b[1;32m   2073\u001b[0m                     \u001b[0morientation\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0morientation\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2074\u001b[0m                     \u001b[0mbbox_inches_restore\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0m_bbox_inches_restore\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2075\u001b[0;31m                     **kwargs)\n\u001b[0m\u001b[1;32m   2076\u001b[0m             \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2077\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mbbox_inches\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mrestore_bbox\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/backends/backend_agg.py\u001b[0m in \u001b[0;36mprint_raw\u001b[0;34m(self, filename_or_obj, *args, **kwargs)\u001b[0m\n\u001b[1;32m    459\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    460\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mprint_raw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfilename_or_obj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 461\u001b[0;31m         \u001b[0mFigureCanvasAgg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdraw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    462\u001b[0m         \u001b[0mrenderer\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_renderer\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    463\u001b[0m         \u001b[0;32mwith\u001b[0m \u001b[0mcbook\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_setattr_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrenderer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdpi\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfigure\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdpi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;31m \u001b[0m\u001b[0;31m\\\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/backends/backend_agg.py\u001b[0m in \u001b[0;36mdraw\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    400\u001b[0m         \u001b[0mtoolbar\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtoolbar\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    401\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 402\u001b[0;31m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfigure\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdraw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrenderer\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    403\u001b[0m             \u001b[0;31m# A GUI class may be need to update a window using this draw, so\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    404\u001b[0m             \u001b[0;31m# don't forget to call the superclass.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/figure.py\u001b[0m in \u001b[0;36mdraw\u001b[0;34m(self, renderer)\u001b[0m\n\u001b[1;32m   1647\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1648\u001b[0m             mimage._draw_list_compositing_images(\n\u001b[0;32m-> 1649\u001b[0;31m                 renderer, self, artists, self.suppressComposite)\n\u001b[0m\u001b[1;32m   1650\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1651\u001b[0m             \u001b[0mrenderer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclose_group\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'figure'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py\u001b[0m in \u001b[0;36mdraw_wrapper\u001b[0;34m(artist, renderer, *args, **kwargs)\u001b[0m\n\u001b[1;32m     48\u001b[0m                 \u001b[0mrenderer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstart_filter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     49\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 50\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mdraw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0martist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrenderer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     51\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     52\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0martist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_agg_filter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/axes/_base.py\u001b[0m in \u001b[0;36mdraw\u001b[0;34m(self, renderer, inframe)\u001b[0m\n\u001b[1;32m   2626\u001b[0m             \u001b[0mrenderer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstop_rasterizing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2627\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2628\u001b[0;31m         \u001b[0mmimage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_draw_list_compositing_images\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrenderer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0martists\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2629\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2630\u001b[0m         \u001b[0mrenderer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclose_group\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'axes'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py\u001b[0m in \u001b[0;36mdraw_wrapper\u001b[0;34m(artist, renderer, *args, **kwargs)\u001b[0m\n\u001b[1;32m     48\u001b[0m                 \u001b[0mrenderer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstart_filter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     49\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 50\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mdraw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0martist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrenderer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     51\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     52\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0martist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_agg_filter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/text.py\u001b[0m in \u001b[0;36mdraw\u001b[0;34m(self, renderer)\u001b[0m\n\u001b[1;32m   2391\u001b[0m         \u001b[0;31m# Draw text, including FancyBboxPatch, after FancyArrowPatch.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2392\u001b[0m         \u001b[0;31m# Otherwise, a wedge arrowstyle can land partly on top of the Bbox.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2393\u001b[0;31m         \u001b[0mText\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdraw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrenderer\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2394\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2395\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mget_window_extent\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrenderer\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/artist.py\u001b[0m in \u001b[0;36mdraw_wrapper\u001b[0;34m(artist, renderer, *args, **kwargs)\u001b[0m\n\u001b[1;32m     48\u001b[0m                 \u001b[0mrenderer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstart_filter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     49\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 50\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mdraw\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0martist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrenderer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     51\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     52\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0martist\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_agg_filter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/text.py\u001b[0m in \u001b[0;36mdraw\u001b[0;34m(self, renderer)\u001b[0m\n\u001b[1;32m    752\u001b[0m                     textrenderer.draw_tex(gc, x, y, clean_line,\n\u001b[1;32m    753\u001b[0m                                           \u001b[0mtextobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_fontproperties\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mangle\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 754\u001b[0;31m                                           mtext=mtext)\n\u001b[0m\u001b[1;32m    755\u001b[0m                 \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    756\u001b[0m                     textrenderer.draw_text(gc, x, y, clean_line,\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/backends/backend_agg.py\u001b[0m in \u001b[0;36mdraw_tex\u001b[0;34m(self, gc, x, y, s, prop, angle, ismath, mtext)\u001b[0m\n\u001b[1;32m    231\u001b[0m         \u001b[0mtexmanager\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_texmanager\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    232\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 233\u001b[0;31m         \u001b[0mZ\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtexmanager\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_grey\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msize\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdpi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    234\u001b[0m         \u001b[0mZ\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mZ\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;36m255.0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0muint8\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    235\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/texmanager.py\u001b[0m in \u001b[0;36mget_grey\u001b[0;34m(self, tex, fontsize, dpi)\u001b[0m\n\u001b[1;32m    418\u001b[0m         \u001b[0malpha\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgrey_arrayd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    419\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0malpha\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 420\u001b[0;31m             \u001b[0mpngfile\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_png\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtex\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfontsize\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdpi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    421\u001b[0m             \u001b[0mX\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_png\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mread_png\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mos\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpath\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjoin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtexcache\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpngfile\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    422\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgrey_arrayd\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0malpha\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mX\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/texmanager.py\u001b[0m in \u001b[0;36mmake_png\u001b[0;34m(self, tex, fontsize, dpi)\u001b[0m\n\u001b[1;32m    383\u001b[0m             self._run_checked_subprocess(\n\u001b[1;32m    384\u001b[0m                 [\"dvipng\", \"-bg\", \"Transparent\", \"-D\", str(dpi),\n\u001b[0;32m--> 385\u001b[0;31m                  \"-T\", \"tight\", \"-o\", pngfile, dvifile], tex)\n\u001b[0m\u001b[1;32m    386\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mpngfile\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    387\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/site-packages/matplotlib/texmanager.py\u001b[0m in \u001b[0;36m_run_checked_subprocess\u001b[0;34m(self, command, tex)\u001b[0m\n\u001b[1;32m    296\u001b[0m             report = subprocess.check_output(command,\n\u001b[1;32m    297\u001b[0m                                              \u001b[0mcwd\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtexcache\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 298\u001b[0;31m                                              stderr=subprocess.STDOUT)\n\u001b[0m\u001b[1;32m    299\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0msubprocess\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mCalledProcessError\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mexc\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    300\u001b[0m             raise RuntimeError(\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/subprocess.py\u001b[0m in \u001b[0;36mcheck_output\u001b[0;34m(timeout, *popenargs, **kwargs)\u001b[0m\n\u001b[1;32m    354\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    355\u001b[0m     return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,\n\u001b[0;32m--> 356\u001b[0;31m                **kwargs).stdout\n\u001b[0m\u001b[1;32m    357\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    358\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/subprocess.py\u001b[0m in \u001b[0;36mrun\u001b[0;34m(input, timeout, check, *popenargs, **kwargs)\u001b[0m\n\u001b[1;32m    423\u001b[0m     \u001b[0;32mwith\u001b[0m \u001b[0mPopen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpopenargs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mprocess\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    424\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 425\u001b[0;31m             \u001b[0mstdout\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mstderr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mprocess\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcommunicate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minput\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtimeout\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtimeout\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    426\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mTimeoutExpired\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    427\u001b[0m             \u001b[0mprocess\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkill\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/anaconda3/lib/python3.6/subprocess.py\u001b[0m in \u001b[0;36mcommunicate\u001b[0;34m(self, input, timeout)\u001b[0m\n\u001b[1;32m    848\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_stdin_write\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0minput\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstdout\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 850\u001b[0;31m                 \u001b[0mstdout\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstdout\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mread\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    851\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstdout\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mclose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    852\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstderr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "ani.save('__k-scheme_' + str(k)+'.mp4',fps=24,dpi=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "CSS style adapted from https://github.com/barbagroup/CFDPython. Copyright (c) Barba group\n",
       "<link href='http://fonts.googleapis.com/css?family=Merriweather' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Bitter' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Oxygen' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Lora' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
       "\n",
       "<style>\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "    }\n",
       "\n",
       "    /*div.cell{\n",
       "        width:800px;\n",
       "        margin-left:16% !important;\n",
       "        margin-right:auto;\n",
       "    } */\n",
       "\n",
       "    /* set the font size in tables */\n",
       "    tr, td, th{\n",
       "        font-size:110%;\n",
       "    }\n",
       "\n",
       "    /* spec for headers */\n",
       "    h1 {\n",
       "        font-family: 'Bitter', serif;\n",
       "    }\n",
       "\n",
       "    h2 {\n",
       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "    \n",
       "    h3{\n",
       "        font-family: 'Fenix', serif;\n",
       "        margin-top:12px;\n",
       "        margin-bottom: 3px;\n",
       "    }\n",
       "\n",
       "    h4{\n",
       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "\n",
       "    h5 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\n",
       "\n",
       "    div.text_cell_render{\n",
       "        font-family: 'Merriweather','Alegreya Sans','Lora', 'Oxygen', \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "        line-height: 160%;\n",
       "        font-size: 130%;\n",
       "    }\n",
       "\n",
       "    .CodeMirror{\n",
       "        font-family: \"Source Code Pro\";\n",
       "        font-size: 100%;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h1 {\n",
       "        font-weight: 200;\n",
       "        font-size: 32pt;\n",
       "        line-height: 120%;\n",
       "        color:#CD2305;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h2 {\n",
       "        font-size: 26pt;\n",
       "        text-align: center;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h3 {\n",
       "        font-size: 20pt;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h4 {\n",
       "        font-size: 18pt;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h5 {\n",
       "        font-weight: 300;\n",
       "        font-size: 16pt;\n",
       "        color: #CD2305;\n",
       "        font-style: italic;\n",
       "        margin-bottom: .5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\n",
       "    \n",
       "    .warning{\n",
       "        color: rgb( 240, 20, 20 )\n",
       "    }  \n",
       "\n",
       "/*  div#notebook {background-color: #1e1e1e; border-top: none;}\n",
       "    div#notebook-container {background-color: rgb(180, 180, 180);}\n",
       " */\n",
       "\n",
       "</style>\n",
       "\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                TeX: {\n",
       "                    extensions: [\"AMSmath.js\"]\n",
       "                },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    availableFonts: [\"TeX\"],\n",
       "                    scale: 100,\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import urllib\n",
    "import requests\n",
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = requests.get(\"https://raw.githubusercontent.com/saadtony/NumericalMethods/master/styles/custom.css\")\n",
    "    return HTML(styles.text)\n",
    "css_styling()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.9"
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 },
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 "nbformat_minor": 2
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