CH_EN_6355_Utah_CFD_Saad/TVD Formulation.ipynb

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    "# Advection using TVD Flux Limiters\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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    "variables": {
     "1 + k": {},
     "1 - k": {}
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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",
    "Our TVD formulation is based on a finite volume discretization with face values for the dependent variable given by:\n",
    "\\begin{equation}\n",
    "{\\phi _f} = {\\phi _{\\rm{C}}} + {1 \\over 2}\\psi ({r_f})\\left( {{\\phi _{\\rm{D}}} - {\\phi _{\\rm{C}}}} \\right)\n",
    "\\end{equation}\n",
    "where $\\psi$ is the limiter function."
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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 = 256 # 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": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "dt= 0.001953125\n",
      "dx= 0.00390625\n"
     ]
    }
   ],
   "source": [
    "cfl = 0.5\n",
    "c = 1.0\n",
    "dt = cfl*dx/abs(c)\n",
    "print('dt=',dt)\n",
    "print('dx=',dx)\n",
    "# μ = 0.001\n",
    "# dt = 0.02"
   ]
  },
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   "cell_type": "code",
   "execution_count": 16,
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    "# finite volume implementation with arrays for fluxes\n",
    "t = 0\n",
    "tend = L/abs(c)\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",
    "schemename='Bounded CD'\n",
    "\n",
    "i = 0\n",
    "while t < tend:    \n",
    "    phin = sol[-1]\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",
    "\n",
    "    if (i%2==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",
    "    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",
    "    # compute r - ratio of successive gradients\n",
    "    numerator = ϕc - ϕu    \n",
    "    denominator = ϕd - ϕc\n",
    "    # find the locations where the denominator is > 1e-12 - that's where we will do the divisions. otherwise, set phi to 1\n",
    "    divideloc = np.where(np.abs(denominator) > 1e-6)\n",
    "    i1 = np.where(numerator < 1e-12)\n",
    "    r = np.ones_like(ϕc)\n",
    "    r[divideloc] = numerator[divideloc]/denominator[divideloc]\n",
    "\n",
    "    # compute face values - these actually live on the faces\n",
    "    ϕf = np.zeros_like(ϕc)\n",
    "    \n",
    "#     # bounded CD    \n",
    "#     schemename='Bounded CD'\n",
    "#     ϕftilde = 0.5 + 0.5*ϕctilde\n",
    "    \n",
    "#     # minmod\n",
    "#     schemename='MinMod'\n",
    "#     psi = np.maximum(0,np.minimum(1,r))\n",
    "    \n",
    "    # SUPERBEE\n",
    "    schemename='Superbee'\n",
    "    a1 = np.maximum(0, np.minimum(1.0,2.0*r))\n",
    "    psi = np.maximum(a1, np.minimum(2.0,r))\n",
    "\n",
    "#     # MUSCL\n",
    "#     schemename='MUSCL'\n",
    "#     psi = np.maximum(0.0, np.minimum(np.minimum(2*r, (r+1)/2.0), 2.0))\n",
    "\n",
    "    # now retrieve phif from phiftilde\n",
    "    ϕf = ϕc + 0.5*psi*(ϕd - ϕc)\n",
    "    f[2:-2] = ϕf\n",
    "    f = c*f # multiply the flux by the velocity\n",
    "    # advect\n",
    "    phi[2:-2] = phin[2:-2] - 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(schemename, 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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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       "      <use transform=\"scale(0.996264)\" xlink:href=\"#CMR17-111\"/>\n",
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       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pafc839ec8a\">\n",
       "   <rect height=\"217.44\" width=\"334.8\" x=\"26.930079\" y=\"10.7\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(sol[0], label='initial condition')\n",
    "plt.plot(sol[-1], label='one residence time')\n",
    "plt.legend()\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     },
     "execution_count": 18,
     "metadata": {},
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   "source": [
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Create Animation in Moving Reference Frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "done!\n"
     ]
    },
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      "text/plain": [
       "<Figure size 600x450 with 1 Axes>"
      ]
     },
     "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,color='0.9')\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",
    "\n",
    "ann = ax.annotate('time ='+str(round(t,4))+' s.' + '\\t'+ schemename, xy=(0,0), xytext=(40, 200),xycoords='figure points')\n",
    "plt.tight_layout()\n",
    "# plt.locator_params(axis='x', nbins=15)\n",
    "\n",
    "# def animate(i):\n",
    "# #     xt = x - i*c*dt/dx\n",
    "# #     line.set_xdata(xt)\n",
    "# #     ax.axes.set_xlim(xt[0],xt[-1])\n",
    "# #     ax.grid()\n",
    "# #     line.set_ydata(np.sin(xt))  # update the data\n",
    "# #     line.set_ydata(sol[i])\n",
    "#     print('time=',i*dt)\n",
    "#     shift =int(np.ceil(i*c*dt/dx))\n",
    "#     f = sol[i]\n",
    "#     line.set_ydata(np.roll(f, -shift))\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",
    "#     return line,\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(-0.1,1.1*max(f))\n",
    "    ann.set_text('time ='+str(round(t,4))+'s (' + str(i)+ ').\\t' + schemename)\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(1.0/cfl)), init_func=init,\n",
    "                              interval=20, blit=False)\n",
    "print('done!')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
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      "time= 0.921875\n",
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      "time= 0.9453125\n",
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      "time= 0.9921875\n",
      "time= 1.0\n"
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      "time= 0.0\n",
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      "time= 0.6484375\n",
      "time= 0.65625\n",
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      "time= 0.953125\n",
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      "time= 0.9765625\n",
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      "time= 0.9921875\n",
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   "source": [
    "ani.save(schemename+'_TVD.gif', writer='imagemagick',fps=24,dpi=200)"
   ]
  },
  {
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   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
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       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
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       "        color: rgb( 240, 20, 20 )\n",
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       "\n",
       "/*  div#notebook {background-color: #1e1e1e; border-top: none;}\n",
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       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
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       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
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      ]
     },
     "execution_count": 11,
     "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()"
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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