CH_EN_6355_Utah_CFD_Saad/Using Sparse Solvers in Python.ipynb

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    "# Using Sparse Solvers in Python\n",
    "## ChEN 6355 - Computational Fluid Dynamics\n",
    "### Tony Saad <br/> Assistant Professor of Chemical Engineering <br/> University of Utah"
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    "We will solve the Poisson equation on the unit square with Dirichlet boundary conditions\n",
    "\\begin{equation}\n",
    "\\nabla^2 u = -S;\\quad u(0) = 0,\\quad u(1) = 0\n",
    "\\end{equation}\n",
    "We will use a random number generator for the RHS."
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   "cell_type": "code",
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   "source": [
    "import numpy as np\n",
    "import scipy.linalg\n",
    "import scipy.sparse\n",
    "import scipy.sparse.linalg\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import pyamg # make sure you install pyamg"
   ]
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      "[[-2178.  1089.     0. ...     0.     0.     0.]\n",
      " [ 1089. -2178.  1089. ...     0.     0.     0.]\n",
      " [    0.  1089. -2178. ...     0.     0.     0.]\n",
      " ...\n",
      " [    0.     0.     0. ... -2178.  1089.     0.]\n",
      " [    0.     0.     0. ...  1089. -2178.  1089.]\n",
      " [    0.     0.     0. ...     0.  1089. -2178.]]\n"
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    "# 1d example\n",
    "l = 1.0\n",
    "ntotal = 34\n",
    "dx = dy = 1/(ntotal -1)\n",
    "\n",
    "n = 32 # interior points excludes BC points where the BC is specified\n",
    "S = -np.random.rand(n)\n",
    "\n",
    "Ae = 1/dx/dx*np.ones(n)\n",
    "Aw = 1/dx/dx*np.ones(n)\n",
    "Ap = -(Ae + Aw) # create Ap\n",
    "\n",
    "# set BCs on east and west coefs\n",
    "Ae[-1] = 0.0 # right wall\n",
    "Aw[0] = 0.0 # left wall\n",
    "\n",
    "# create the diagonals\n",
    "d0 = Ap\n",
    "de = Ae[:-1]\n",
    "dw = Aw[1:]\n",
    "\n",
    "A = scipy.sparse.diags([d0, de, dw], [0, 1, -1], format='csr')\n",
    "print(A.toarray())"
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    "# direct solver\n",
    "Ainv = scipy.sparse.linalg.inv(A.tocsc())\n",
    "u = Ainv@S\n",
    "plt.plot(u)\n",
    "# to add the values at the boundary points you can augment u with two extra values"
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   "execution_count": 39,
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   ],
   "source": [
    "# scipy sparse solver\n",
    "u = scipy.sparse.linalg.spsolve(A,S)\n",
    "plt.plot(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
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    "# cg solver\n",
    "u,err = scipy.sparse.linalg.cg(A,S)\n",
    "plt.plot(u)"
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  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
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   ],
   "source": [
    "# amg solver\n",
    "ml = pyamg.ruge_stuben_solver(A)                   \n",
    "u = ml.solve(S, tol=1e-9,cycle='V')\n",
    "plt.plot(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
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       "<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": 42,
     "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()\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}