hw3_BVP_2_source.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import print_function\n",
    "%matplotlib inline\n",
    "%precision 8\n",
    "import numpy\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "h1",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "Before you turn this problem in, make sure everything runs as expected. First, restart the kernel (in the menubar, select Kernel $\\rightarrow$ Restart) and then run all cells (in the menubar, select Cell $\\rightarrow$ Run All).\n",
    "\n",
    "Make sure you fill in any place that says YOUR CODE HERE or \"YOUR ANSWER HERE\", as well as your name and collaborators below:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "h2",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "# HW 3: BVP Problems II"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q1",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "## Question 1\n",
    "\n",
    "Consider the two-dimensional Poisson problem defined as\n",
    "$$\n",
    "    \\nabla^2 u = f(x,y) ~~~~ \\Omega = [0, 1] \\times [0, 1]\n",
    "$$\n",
    "with Dirichlet boundary conditions $u(x,y)|_{\\partial \\Omega} = 0$.\n",
    "\n",
    "Note that the notation $\\partial \\Omega$ often refers to the boundary of $\\Omega$ and $u(x,y)|_{\\partial \\Omega}$ the solution evaluted at the boundary."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q1-a",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(a)** [5] If we wanted to consider solutions of the form\n",
    "$$\n",
    "    u(x,y) = (y^3 - y) (\\cos(2 \\pi x) - 1)\n",
    "$$\n",
    "what should we require $f(x,y)$ to be?  Is this consistent with the boundary conditions?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A1-a",
     "locked": false,
     "points": 5,
     "schema_version": 3,
     "solution": true
    }
   },
   "source": [
    "The derivatives we need are\n",
    "$$\n",
    "    u_{xx} = -4 \\pi^2 (y^3 - y) \\cos(2 \\pi x)\n",
    "$$\n",
    "and\n",
    "$$\n",
    "    u_{yy} = 6 y (\\cos(2 \\pi x) - 1)\n",
    "$$\n",
    "leading us to the function\n",
    "$$\n",
    "    f(x,y) = - 4 \\pi^2 (y^3 - y) \\cos(2\\pi x) + 6 y (\\cos(2\\pi x) - 1)\n",
    "$$\n",
    "\n",
    "Since assumed solution $u(x,y) = 0$ along the boundaries specified are zero these are consistent with the boundary conditions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q1-b",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(b)** [10] Implement a finite difference method for the problem above with grid spacing $\\Delta x = \\Delta y$ using a 9-point Laplacian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "A1-b",
     "locked": false,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [],
   "source": [
    "# Suggested modules to use to construct matrix A\n",
    "# You do not have to use these, they just may be helpful\n",
    "import scipy.sparse as sparse\n",
    "import scipy.sparse.linalg as linalg\n",
    "\n",
    "def solve_9point(m, f):\n",
    "    ### BEGIN SOLUTION\n",
    "    x = numpy.linspace(0, 1.0, m + 2)\n",
    "    y = numpy.linspace(0, 1.0, m + 2)\n",
    "    delta_x = 1.0 / (m + 1)\n",
    "\n",
    "    # Construct A\n",
    "    e = numpy.ones(m)\n",
    "    T = sparse.spdiags([4.0 * e, -20.0 * e, 4.0 * e], [-1, 0, 1], m, m)\n",
    "    I = sparse.eye(m)\n",
    "    S = sparse.spdiags([e, e], [-1, 1], m, m)\n",
    "    OI = sparse.spdiags([e, 4.0 * e, e], [-1, 0, 1], m, m)\n",
    "    A = (sparse.kron(I, T) + sparse.kron(S, OI)).tocsr()\n",
    "    A /= 6.0 * delta_x**2\n",
    "    \n",
    "    # Right-hand-side\n",
    "    X, Y = numpy.meshgrid(x[1:-1], y[1:-1])\n",
    "    b = f(X, Y).reshape(m**2, order='F')\n",
    "\n",
    "    # Solve\n",
    "    U = numpy.zeros((m+2, m+2))\n",
    "    U[1:-1, 1:-1] = linalg.spsolve(A, b).reshape((m, m), order='F')\n",
    "\n",
    "    ### END SOLUTION\n",
    "    return x, y, U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "T1-b",
     "locked": true,
     "points": 10,
     "schema_version": 3,
     "solution": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00026543132573227964\n",
      "Success!\n"
     ]
    }
   ],
   "source": [
    "f = lambda x, y: -(20.0 * y**3 + 9.0 * numpy.pi**2 * (y - y**5)) * numpy.sin(3.0 * numpy.pi * x)\n",
    "u_true = lambda x, y: (y - y**5) * numpy.sin(3.0 * numpy.pi * x)\n",
    "\n",
    "x, y, U = solve_9point(100, f)\n",
    "X, Y = numpy.meshgrid(x, y)\n",
    "error = numpy.linalg.norm((x[1] - x[0]) * (u_true(X, Y) - U), ord=1)\n",
    "print(error)\n",
    "assert error < 1e-3\n",
    "### THIS IS SPARTA!\n",
    "print(\"Success!\")\n",
    "### UPS... THIS IS NOT SPARTA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q1-c",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(c)** [5] Show that the method is second-order accurate by preforming a convergence study (i.e. plot the error vs. $\\Delta x$ and compare this to the slopes for first and second order accurate methods)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A1-c",
     "locked": false,
     "points": 5,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [
    {
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4I/Q5FaZOhYsvhrPOgjLF2AzwFkeCyc3NZcGCBRxzzDE88MADZGdns3r1aqpWrUpOTk6Bj+nfvz+DBg0iMzMTsJlS9913H9dff/12961RowY1atTg66+/BthmbvdJJ53EkCFD2LhxIwC//vora9asifErdM4V6KuvYO+9oXNnePNNOO44ePdd2GcfO5+SwoaK1bnjDjjsMMjKgrffhhdeKP5QvcURonPOOYdx48axbNky6tWrx1133cUFF1zAeeedR3Z2NqrKVVddRY0aNejatSunn346o0aN4qmnnqJDhw5brtO6dWseeOABunbtysaNGylfvjwPPvggrVu3LvB5X3rpJS6++GJEZMsYBsAll1xCZmYmbdq0QVWpU6dOkaf1OueisGYNfPyxtSxOOsnGSps1g27dbLz0hBOgYsXtHjZ2LNx9N5x/vjVAatUq/tChlOw53rZtW82/H8fs2bNp2rRpSBElL39fnSvEW29ZK+Ljj2HdOqhTx2ZHXXPNDh+yZg1MnmwNELAJVYcdFv9QRWSaqrYt6FxSd1Ul+joO51ySW74cPv106+0nn4RJk2xgYuxYWLy40KTx3//CIYfAqafacAcUT9LYmaTuqlLVD4AP2rZte2nYsTjnSok//7SB7eHD4csvbdR66VKoXh1GjLBWxk5GsleuhP79YdgwaNLEck+dAjdxDUdSJw7nnCtWL79srQlVOPBAuOEG6NULqlWz83kD3YVYvdpaGX/8ATfeCHfcAXvuGd+wo+WJwznndsW8edaqGD7c/sL37AlHHQW3324D3M2bQxSFP9esgcqVoUoVGDDALpUI3VIFSeoxDueci6l//oH0dGjTBg44wFoUmzbZKm6wfqU774QWLYqcNFThtdegYUObkQtw1VWJmzTAE4dzzu2YKnz/vY1ZAJQvDy++aFNlH37YWh1Tp9o02l0wfz6ccgpccIH1bBWhJysheOIISd4iv2bNmtG8eXOeeOKJqK/RqVMn8k8zBqtLdc0119C4cWOaNGlC9+7dWbhwYZGueeedd/Lwww9HHYtzSUPVSn3ceKO1IA49FC67DHJzrRUxYwZMnAjXXw+7UYZn2DDrzRo/3iZbTZgABx8cw9cRR544QlKuXDkeeeQRZs2axeTJkxk8eDCzZs2KybVvueUWcnJy+OWXX5gzZw49evSgV69e5F+zo6rk5ubu1nN5mXWXFHJz7Qtg4EBo3x4efdS6o5591pJF3kyoSpVi8pSrV8ORR8LMmVZnKq+3qyTwxBGSfffdlzZt2gBQtWpVmjZtyqJFiwBrSdx44420a9eOAw88cEsZ9XXr1nH22WfTtGlTevbsybp167a77tq1a3nppZd47LHHKBv8JF500UVUqFCBsWPHkpmZyUEHHcQFF1xAixYtWLBgAenp6Rx44IEcffTR/PLLL1uu9dtvv9G5c2cOO+wwOnTowM8//wzYPiJ9+/alffv23HDDDXF9n5yLm02bbKHEFVdA3bq2yg6s8NPLL8OSJTYPNi3NSoHE4OkefNDWAIIli08/tTJUJY3PqgqEWVU9MzOT7777jvbt2285tmnTJv73v/8xZswY7rrrLr744guGDBlCpUqVmD17Nj/++OOWxBNp7ty5NGjQgGp50/8Cbdu25aeffuKAAw5gzpw5vPLKKxxxxBFMmzaNt956i++//55NmzbRpk0bDgtG5dLS0hg6dChNmjTh22+/5YorrmDs2LEALFy4kIkTJ25JTs6VGMuWWTfUqFG2QK9SJfsFzyvx0aqVfcXQ999Dnz4wfbpVQj/77OItShhrnjhCtnr1ak477TQef/zxbf7Y9+rVC4DDDjtsS/HC8ePHc9VVVwFWer1ly5a79JwNGzbkiCOOAGDChAn07NmTSkHzu1swyLd69WomTpzIGWecseVxGzZs2PL9GWec4UnDlQzr1tlH+02b7BNe1arwxRdWTPC006xWVIy6n/Jbvx7uuQceeMCKaL/3nj1lSZfUiUNEugJdGzduvNP7hlFVfePGjZx22mmkpqZuSRR5KlSoAEDZsmWjGkc44IADmD9/Pjk5OVStWnXL8WnTpnHqqacCRSuxnpubS40aNfj+++8LPO9l1l1Cy8mBMWNsjcWYMbZI4ogjLHFUqGClyovhI//YsXDffdC7tw2ZFLDhZ4lUghtLO5fIW8eqKn369KFp06Zcd911RXpMx44deeONNwCYOXNmgXuKV65cmd69e3PdddexefNmAF599VXWrl3LscceW+A1R44cybp168jJyeGDDz4AoFq1ajRq1Ih33313S7w//PDDLr1W54pF5LYDl15q/UFffQXnnWc7HI0fv/V8HJPG6tX2dGA9YNOn25BJsiQNSPLEkci++eYbXnvtNcaOHUvr1q1p3bo1Y8aMKfQxl19+OatXr6Zp06bcfvvtW8Yi8hs0aBAVK1bkwAMPpEmTJrz77ruMGDECKWBBUps2bTjrrLNo1aoVXbp04fDDD99yLiMjg2HDhtGqVSuaN2/OqFGjdu9FOxdrS5fahhRduliN8Xnz7Hj//pY0Fi+GoUPh+ONtDUacffaZrf3r3n1rUcJDD4370xY7L6vuYsrfV1csfv7ZZq589ZVNo91/fxs8uOoqqFev2MNZsQKuu85aFgcdZGs0jjqq2MOIqcLKqif1GIdzLklkZcH779vc1Z49rVTs0qVw8802btGqVVR1oWIpryjhn3/a1hq33VbgHkxJxROHcy4xzZ1rg9vvvWdlPcDmsvbsad1SM2aEGt7q1VaQsEoVuOkma2EkY7dUQXyMwzmXGFStZZHnwgvtL7KIzWedMweefz608PKowiuvbFuUsF+/0pM0oJS3OFS1wAFjt2tKw3iZizFV+O67rS2L33+3Fds1asBTT1nLokGDsKPcIivLylZ9+ikcfTTsu2/YEYWj1CaOihUrsnz5cmrVquXJIwZUleXLl1Mx2Tt3XeyMHWvLqTMzrVDTv/8NV1+9tWhTgn2Ef/55uPZaawA9/TRcfnnJXv29O0pt4qhXrx4LFy5kad6cObfbKlasSL0QZrS4EmDzZvj6a2tZdOliX/XqQdOmcOutNn+1du2woyzUunXWynj2WeumKs1K7XRc51ycqVppj/fes/0sliyx6Ub33mslyRPcxo225UbDhnDuuVurqpeWDorCpuOW0oaWcy4uNmyAyAoDl18Ob7wBxxwD77xjU2hLQNKYPh3atbPptXkD4GXKlJ6ksTOltqvKORcja9bAJ59YN9SHH0K5cvDXX7ZS+4MPbO3FnnuGHWWRrFsHd98NDz1kS0Xef99m/7pteeJwzu26YcNsY4l162wG1Jln2gruvI/mJayKwLhxcP/9cPHF1k21115hR5SYPHE454rm779h9GhrWdx8M/zrX1aY6aKLLFl07GitjRImJ8d2gj3pJBuz//77mG/HkXRK3v+yc674rF9vq92GD4cvv7Q9LRo0sIFusC1WIzYgK2k++cTWZSxZAvPnW/eUJ42dS+rBcRHpKiLPZWdnhx2KcyXHokUwaZJ9X6aMrd7OzLSKs1Om2Pc9eoQY4O5bvtz2yOjSBSpXtiUldeqEHVXJ4dNxnXO2Ynv4cPuaPBmaNIFffrGxisWLbYl0kkwpWr3aKtguWWI58dZbbW8nty2vjuuc27Ebb4QHH7TvDz0U0tO3HeDeb7/wYouhnBzbNbZKFRg40IoSerfUrvHE4VxpoWoVZd97z1oWI0day+LEE62fplcv29ciyajCSy9ZT9vw4bak5Iorwo6qZEvqMQ7nHLbo7qab4MAD7SN2erolilWr7Pxxx9lf1RKYNDIyMkhJSaFMmTKkpKSQkZGxzfl58ywv9ulje2Z4RZzY8BaHc8kmN9cGtzdv3jpF9sknoUMHGDDABrb33jvsKHdbRkYGaWlprF27FoCsrCzS0tIASE1NZehQW6RetiwMGQJpaaW3KGGs+eC4c8lg0yYYP976YkaMgD/+sD6ZsWPt/Jo1Nn0oiaSkpJAVuX9HoGHDhmRmZvLUUzbdduhQqF8/hABLuMIGxz1xOFdSbd68tQR5t25W3mPPPeHkk21w+5RToFq1cGOMozJlyuTbA6Y8cAPwO6oZpa4oYaz5rCrnksW6dbaL0PDh9nH6l1+gZk34z39sYULnzknXstiRBg0aRLQ4DgOGAa2oUuU1wLul4snfWudKgtmz4ayzbFC7Z0/46CNrUaxZY+dPOslaGaUkaQCkp6ez5541gfuBb4Ha7LHHGQwd6n/W4s3fYecSUXY2vP66bX4EVml23Dg47zz47DOrPvvyy6W68z41NZV+/d4DbgReon79Lrz4Yg9SU1PDDi3p+RiHc4li2TIYNcq6ob74wnYSuuQS27MUbLaU97+wahV8842VCwFbmnLIIeHGlIx8jMO5RLV6tS1lBttze9YsaNTI9t4+7TTbTSiPJw0++gj69rVaU/Pn226znjSKnycO54rb/Pm2Q9Dw4ZYo/vgD9tgDHnvM/hIeeqhPBcpn2TK45hrIyIDmzW3xe4JvUZ7UPHE4V1zGjrUV3FOm2O1DDrGWxYYNljhOPDHc+BLU6tXQsqUljzvusO1c99gj7KhKN08czsXLrFn20fjkk6FtWyvBqgqDBlk3VJMmYUeY0FatsmUoVarA7bdbUULvlkoMnjicixVV2z4urzz5zz9bl1OVKpY4jjpqa2vD7ZAqvPCCVUd5/3049lgb13CJwxOHc7tD1farqFvXVnIffzysXAmdOtle3D172l4Wrkh++w0uvdQ2G+zUyTYbdInHE4dz0dq82eaDDh9uH4krVIA5c6yY4IgR0KyZj9zugiFDrChh+fLw7LM2E9knkiUmTxzOReOll+Dmm20BXoUKVuLjtNNsjUXZslaN1u2S3Fyr8D5kiJc/T3Sez53bkQ0b4MMP4aKL4Ndf7Vjt2pYc3nrL9rkYORLOP39rsUFXZP/8A3fdZQvkwTZXGj3ak0ZJUKQWh4gIUE9VF8Q5HufCtWGDrTIbPtySRt7Unh49bCOkrl3ty+2W//3PNleaOdPqM553ni9dKUmK1OJQq0syJs6xOBeOnJytLYp16+Dss60C7RlnWBJZsgS6dw83xiSxdq2NYxx5pM0h+PBDePrpsKNy0YpmjGO6iByuqqHOJxSRHsApQDVgmKp+FmY8roRascL6RYYPt6KBbdtaQcEaNezjcIsWNtjtYmr8eHj0UZte+8ADSb1dSFKL5jejPZAqIlnAGkCwxkjLol5ARF4ETgWWqGqLiOOdgSeAssALqnr/jq6hqiOBkSKyF/Aw4InDReemm+CRR2zXvPr14fLL4fTTt55v3Tq00JJRdrbl5FNOsbkEP/1kE89cyRVN4jgpBs/3MvA08GreAREpCwwGTgAWAlNEZDSWRAble/zFqrok+P7W4HHO7diiRTZldsQIeOcdG9xu1Qquu86SRdu23rkeRx98YK2LFSu2FiX0pFHyFTlxqGqWiLQCOgSHJqjqD9E8maqOF5GUfIfbAXNVdR6AiLwFdFfVQVjrZBvBQP39wMeqOj2a53elxPLltlfF8OEwaZIda9p061+uc86xLxc3S5fCVVfZ5LNDDrFq8b60JXkUeTquiFwNZAB7B1+vi8iVMYihLhA5W2thcGxHrgSOB04XkR0WIhCRNBGZKiJTly5dGoMwXUL75RfrAwHrG+nf3wa677nHakbNmgVt2oQbYymRk2PJYvhwuPtumDrVGnYueUTTVdUHaK+qawBE5AFgEvBUPALbEVV9EniyCPd7DngObCOneMflipmqzeXMqws1cyb06mXf778/ZGV5vYpitnKlzS2oWtXWZxx9tJVAd8knmgWAAmyOuL05OLa7FgGR+1/WC445t2Pdulmt7bvvhpo14Ykn7CuPJ41ik5sLQ4dCw4bw3//ascsu86SRzKJpcbwEfCsiI4LbPYBhMYhhCtBERBphCeNs4NwYXNclg9xcmDzZWhL//S98+62V+jjjDJum06MH/N//hR1lqTVnjhUl/OorKxfSqFHYEbniEM3K8XeBccDRweGLVPW7aJ5MRN4EOgG1RWQhcIeqDhORfsCn2EyqF1X1p2iuW8jzdQW6Nm7cOBaXc8Xp11/hySdtNtTixbZzzwkn2G4+devCBReEHWGp9/TTVvq8QgUYNswqs/gEtdJBbFF4Ee4oMkNVS+Q2Km3bttWpU6eGHYYrzD//2A55DRrYfM1Jk+wjbF4RwVNPherVw47SRXjmGVs7+cwzsN9+YUfjYk1EpqlqgdMaokkcrwBPh71yfFd44khQ69fbX57hw20V98qVtofFk09aF9W6dVC5cthRusCGDZCebhsXnn++zU8Ab2Ukq8ISR7GuHHeO3FzbZEHVWha//w577WW1oE47zbqjwO7jSSNhTJ5sRQlnzbLcfv75njBKs2jGONKArPiGE1s+xpEgsrOtmt3w4bbeYuZM+6tz112wzz5wzDG2e49LOGvWwK232oS1evVgzBjo0iXsqFzYfIzDxc9XX8GDD8IXX9gYxn772VqLBx6ASpXCjs4VwSefWKK44goYNMiLEpYmseqqSojquC6B/fmnzYI68UQ44AD4+29bzd2vn9WFat/e9wItAVassKKEXbva3IRZs6xii3N5omlx/Aw0ATIpYWMc3uKIowULrIjge+/ZPtyq8PjjcPXVtjd3mTLeGV6CjBhhrYvs7K2lvVzpFKsWRyyq47pksGaNDVyvWWNTbDZssOJEd9xhA9x5S4Z9O9US46+/bND73XetqvxHH3nScDsWTeKYD6QC+6vq3SLSAPg/EnjA3AfHY2j27K11oSpWtHUWlSvDq6/aX5oDDww7QreLcnKsesvKlTbddsAAn6vgChdNV9UQIBc4VlWbBhspfaaqh8czwFjwrqrd8OqrcP/9ljgA/vUvG6+45hrvgirhVqywmdAAL7xgRQkPPjjcmFziKKyrKpqRyvaq+h9gPYCqrgD2iEF8LlGo2rapN95oGyqALdLbZx+rL7FokY1jXHutJ40SLDcXBg+2ooRffGHHLrnEk4Yrumi6qjYGu/UpgIjUwVogriTbvBkmTrQuqPfft8HucuWgY0crIpiWZl8uKfzyiyWJr7+2tZbei+t2RTQtjieBEcDeIpIOfA3cF5eoXHxt2gRLgh14Fy60JDF0qI1VvPKKnTvllFBDdLH31FO2a+7MmfDSS/Dpp5CSEnZUriSKZuvYDBGZBhyHTcXtoaqz4xaZi60NG6ws+fDhto9nhw4297JhQ5tC06GD7cDjkla5cvZ5YPBgr0Tvdk+RB8dLoohZVZfOmTMn7HDCc9ttVjhw1Spb+tu1K5x9tlWcdUlr/XrbOffAA6F3bxvC8qEpV1SxGhzPu9jxIvK8iLQObidsB7iqfqCqadVLUznunBx4+2047zz7ywE2deb0061lsWQJvP66J40k98031vN4333wXbBrjicNFyvRDI7n6QNcDtwqIjWB1jGNyEVv1SoYOdK6oT791Lql9tkH5s6FFi3guuvCjtAVk9Wr4ZZbbBJcgwb243DiiWFH5ZLNrhQOylHVlaraHzgRSPh1HElp6VIb2AZbY9G7N0yfDn37wvjxNnW2RYtwY3TF7uuvLWlceaUNgnvScPGwKy2Oj/K+UdWbROTKGMbjCrN4sQ1ov/eeJYe0NBgyBNq1s/UXhx3mRQRLob//hgkTbEuTzp3h5599Ib+Lr6gTh6qOyvteRL5R1aNiG5IrUK9eljTASpXecguceabdFoHDveFXGg0fDv/5jw1tZWVZfSlPGi7edvfjqe80HA9z5liZj27dtu7P2a6dTZGZNcu+7rnHCgu6pJORkUFKSgplypQhJSWFjIyM7e7z55823+H0022bk2++8aKErvjstMUhIk8BM4KvmaqaE3E6oefylqgih5mZtvhu+HCYMcOOtWsHy5ZBnTpw002hhueKR0ZGBmlpaaxduxaArKws0oKV+6mpqcDWooSrVtnni+uvtzUazhWXna7jCKbbHgK0BFoAq9iaSNJUtU68g9xdCVnkUNUGs/fZx/bkHDUKevaEo46yj5G9ekH9+mFH6YpZSkoKWVnbF5xu2LAh06ZlUquW3R42zIoSHnRQMQfoSo3C1nFEvQBQROqxNZEcoqrn7X6I8ZUwiSM3FyZP3loXKjMT7rzT9rHYsMHKlfqS3lKtTJkybP87WQboR+XKTzBihNWYci7eYrWREwCquhBYCHy8u4GVKps328fD336zzQ6OP95WdHfrZucrVPCk4WjQoEG+FsfBwAvAUXTo4C0Mlxh87mY8bNwIn31m02XzVmiXLQt9+sBrr9nq7TFj4OKLfUTTbSM9PZ1KlSoFt64BvgcOpm/fiYwZY4v6nAubD6nF0uTJVmV29GjrdqpSxRLHxo3Wyrj55rAjdAkubwB84MCBZGWtp1Klz3nooQ1cccVpIUfm3FaeOHbHmjXwySdWlrxOHfjhBxvk7tbN9t4+8UTbZtW5Ili3Du6+Gw46KJXMzFQvSugSVlGm4+ZQ8LRbAVRVq8U8qkS2ahV8+KENcH/8sf22P/+87Y5zwQVw0UWwh2+M6KIzYYL9CP36q22wCJ40XOLaaeJQVd+kITfXSnn8/betttqwAfbd18YoTjvN9rIA2HPPcON0JU5Oji3ReeYZ21Tp889t3oRziSypu6p2ewHgzJn28a9aNWth1Kxpdarbt4cjj/S6UG63TZxo5cauuQbuvRcqVw47Iud2Lqp1HCKyF9AE2NJxr6rj4xBXTO3yOo7Fi+HYY+Gcc2ythXMxsHy51ajs2dNuz5kDTZqEG5Nz+cVkHYeIXAJcDdTD5ggeAUwCjo1BjIlpv/2s1KhzMaAK774L/frZvIq8ooSeNFxJE01fy9XY3htZqnoMcCiwMh5BOZdsFi+2KjJnnWVrMSZN8iU8ruSKZoxjvaquFxFEpIKq/iwivo7VuZ3IyYFWrWx3vgcftGEzL0roSrJofnwXikgNYCTwuYisALavxuacA6ywce3aULUqPPSQ1a/0bimXDIrcVaWqPYMtY+8EbgOGAd3jFZhzJdXmzfD449Cwoe35DXDhhZ40XPIocuIQkVeCFgeq+hUwAXg2TnE5VyLNmmXlzq+9Fjp1gmbNwo7IudiLZnC8paquzLuhqiuwAXLnHPDoo3DooTa9NiPDCgz4liouGUWTOMoE6zgAEJGaJPkCQueiUbmyzZyaPRvOPddLhrjkFc0f/keASSLybnD7DCA99iE5VzKsXWvrQps2teozaWlw2WVhR+Vc/EUzOP4qcBrwV/DVS1Vfi1dgsSAiXUXkuezs7LBDcUlm3DibYvvww9bCAG9huNIjqmJLqvqTqj4dfM2KV1CxoqofqGpa9erVww7FJYnsbOjbF445xlaCjx1rU22dK012mjhE5Ovg3xwRWRXxlSMiq+IfonOJY/Jkq6J//fXw44+WQJwrbYpSVv1oERGguarOL4aYnEsoy5ZZUcJeveCkk2zW1P77hx2Vc+EpUleVWgndj+Ici3MJRRXeessGvy+4wKragicN56IZ45guIofHLRLnEsiiRdC9u1XU339/66KqVSvsqJxLDNFMx20PnCcimcAatm4d2zIegTkXlryihGvXwiOPwNVXQ9myYUflXOKIJnGcFLconEsAS5dCnTpWlPCRR6wo4a5uHulcMoumq2o+0AHorapZgAL7xCUq54rR5s2WKBo2hE8+sWO9e3vScG5HokkczwBHAucEt3OAwTGPyLliNHOmbR/fvz+ccAIcckjYETmX+KJJHO1V9T/AethS5HCPuETlXDF4+GFo0wYyM2321MiRULdu2FE5l/iiSRwbRaQs1kWFiNQBcuMSlXPFoHp1OPNMK4V+1lleMsS5ooomcTwJjAD2FpF04GvgvrhE5VwcrF1rK75ffNFuX3IJvP667/3tXLSKPKtKVTNEZBpwHDYVt4eqzo5bZM7F0JdfWqKYNw9uuMGOeQvDuV0T1X4aqvoz8HOcYnEu5rKzYcAAqy/VuLFVtf33v8OOyrmSrciJQ0TaAgOBhsHjfAGgS3jffmtdUzfcAHfeCXvuGXZEzpV80bQ4MoABwAxKyKC4iHQFujb2CfmlypIl8NVXcMYZcOKJVpSwUaOwo3IueUQzOL5UVUer6u+qmpX3FbfIYsD34yhdVG2wu2lTuOiirUUJPWk4F1vRtDjuEJEXgP8CG/IOqur7MY/KuSgtWGAbLI0ZA0ccAS+84EUJnYuXaBLHRcDBQHm2dlUp4InDhSonB1q3hvXr4fHHoV8/L0roXDxFkzgOV9WD4haJc1H66y/YZx8rSvjEE1aU0LulnIu/aMY4JopIs7hF4lwRbdpk+3ynpGwtSnjeeZ40nCsu0bQ4jgB+EJF52BiHT8d1xe6HH6BPH5g2DXr0gJb+0+dcsYsmcXSOWxTOFcEDD8Ctt0LNmvDOO3D66b7627kwRJM4eu/g+N2xCMS5nalZE849Fx591GdMORemaMY41kR8bQa6AClxiMk5ANasgWuvtam1AJdeCq+84knDubBFU+TwkcjbIvIw8GnMI3IO+OILSxSZmXDjjWFH45yLFE2LI79KQL1YBeIcwIoVNvh9wgmwxx4wfjzcf3/YUTnnIkVT5HAGwSZOQFmgDnBPPIJypdfUqfDqq3DzzXD77VCxYtgROefyi2Zw/NSI7zcBf6nqphjH40qhv/6yooRnnmktjblzoWHDsKNyzu1INF1VfwJHAalAH+AWEbk9LlG5Ei8jI4OUlBTKlClDSkoKGRkZ291H1VoXTZta99Tff9txTxrOJbZoEscooDvW2oicYeXcNjIyMkhLSyMrKwtVJSsri7S0tG2SR1YWdOkCvXtb4pgyxabbOucSn6jqzu8FiMhMVW0R53jiom3btjp16tSwwyg1UlJSyMravuJ+w4YNyczMZNUqKxfyzz8waBD85z9QZnemaTjnYk5Epqlq24LORTPGMVFEDlHVGTGKyyWp+fPnF3g8K8uq8VerBk89ZUUJU1KKMTDnXExE8znvaGCaiPwiIj+KyAwR+TFegbmSq0GDBvmOlANuBDL5+GM7kprqScO5kiqaFkeXuEXhkkp6ejppaWmsXbsWaA0MA9pw+OHzOfTQ/EnFOVfSRLNyPKG3iXWJIzU1FYB+/RaxcuV1lCmzgn79xvPEEx1Djsw5FwvRtDicK7LU1FQ2bLCV348+WoeaNeuEHZJzLkZ8LouLmdWr4cor4bnn7PbFF8PLL/s0W+eSTYlLHCLSVESGish7InJ52PE48+mn0Lw5DB5sazScc8mrWBOHiLwoIktEZGa+452D2VpzReSmwq6hqrNVtS9wJraS3YXo77/hwguhc2eoVAkmTID09LCjcs7FU3G3OF4m306CIlIWGIzN2moGnCMizUTkEBH5MN/X3sFjugEfAWOKN3yX37RpkJEBAwfCd9/Z2gznXHIr1sFxVR0vIin5DrcD5qrqPAAReQvorqqD2LawYuR1RgOjReQj4I2C7iMiaUAaFLSuwO2OP/+EcePg7LOtKOG8eVC/fthROeeKSyKMcdQFFkTcXhgcK5CIdBKRJ0XkWQppcajqc6raVlXb1qnjM3piQdUGu5s2tU2Wli+34540nCtdStx0XFUdB4wLOYxSJzMT0tLg88+hQwd4/nnfwtW50ioREsciIPIza73gmEsQq1ZBmzawcSM88wxcdpkXJXSuNEuExDEFaCIijbCEcTZwbiwuLCJdga6NGzeOxeVKncWLYb/9rCjh4ME28O3DRc654p6O+yYwCThIRBaKSJ9gF8F+wKfAbOAdVf0pFs+nqh+oalr16tVjcblSY+NGm1LbqBGMCUaRzjnHk4ZzzhT3rKpzdnB8DD61NiFMm2a78f3wg23lethhYUfknEs03lPttkhPh/btYckSGDEC3n4b9tkn7Kicc4nGE4fbYt99bRX4rFnQo0fY0TjnElVSJw4R6Soiz2VnZ4cdSkJatcq2bY0sSvjCC1CjRqhhOecSXFInDh8c37ExY6BFCxgyBBYs2Pn9nXMuTyJMx3XFaNkyuPZaeP11WwE+cSIccUTYUTnnSpKkbnG47X3/vQ1633abFSX0pOGci5a3OEqBxYutKOG558Lxx1tRwnr1wo7KOVdSJXWLo7QPjqvCsGHQrBn07Wt7Z4AnDefc7knqxFGaB8fnzbPWxSWXQOvWMH26b+HqnIsN76pKQqtW2YrvzZth6FArge5FCZ1zseKJI4ksWgR161pRwqFDrSihd0s552LNP4cmgX/+gXvugf3331qU8KyzPGk45+LDWxwl3JQpVpRwxgzbyvXww8OOyDmX7JK6xZHss6ruvdfWYSxfDqNGwZtvgu+S65yLt6ROHMk+q6pePWttzJoF3bqFHY1zrrRI6sSRbLKzbT3G0KF2+8ILrUBhkuZF51yC8sRRQnz4ITRvDs8/D3/8EXY0zrnSzBNHglu61EqFdO0Ke+0FkybBXXeFHZVzrjTzxJHgfvwRhg+3ZDFtGrRrF3ZEzrnSzqfjJqBFi+DLL+G88+C44+D332G//cKOyjnnTFK3OEradFxVG8No1gyuuGJrUUJPGs65RJLUiaMkTcedO9daF2lpVmfqu++8KKFzLjF5V1UCyM6Gtm23tjj69AGRsKNyzrmCeeII0YIFUL++rcN47jkrSli3bthROedc4ZK6qypRbdgAd9wBBxxg6zMAzjzTk4ZzrmTwFkcx+/Zb64r66SebNeV7fjvnShpvcRSju+6CI4+0MY2PPoLXXoPatcOOyjnnouOJoxilpFitqZ9+gpNPDjsa55zbNZ444mjlSpteO2SI3e7dG555xnboc865kiqpE0eYCwBHj7aihMOGwZIlxf70zjkXN0mdOMJYALhkie3E17071Kplg+F33FFsT++cc3GX1IkjDDNnwsiRtgf41Km2sM8555KJT8eNgQULrCjhBRfAscdaUcJ99w07Kueciw9vcexARkYGKSkplClThpSUFDIyMra7T26uDXw3awZXXgkrVthxTxrOuWTmiaMAGRkZpKWlkZWVhaqSlZVFWlraNsljzhw45hirYnvEEfD997bRknPOJTtR1bBjiLu2bdvq1KlTi3z/lJQUsrKytjvesGFDMjMzyc6Ghg2tEOGjj9re316U0DmXTERkmqoWOErrYxwFmD9/foHH83JJ9erwwgtWlNC7pZxzpY13VRWgQYMG+Y7sAdwNzNlSlPD00z1pOOdKJ08cBUhPT6dSpUrBrSOA74DbOProBRx5ZIiBOedcAvDEUYDU1FSee+456tRJA76hbNlqDBgwlgkT9qdWrbCjc865cCX14LiIdAW6Nm7c+NI5c+ZE/fjVq+H+++HGG6Fq1djH55xziaqwwfGkThx5op1V5ZxzpV1hicO7qpxzzkXFE4dzzrmoeOJwzjkXFU8czjnnouKJwznnXFQ8cTjnnIuKJw7nnHNR8cThnHMuKqViAaCILAW2r5Oe+KoD2WEHsRsSJf7ijCMezxXLa+7OtXb1sbvyuNrAsl14Lret3fn/bqiqdQo6USoSR0klIs+palrYceyqRIm/OOOIx3PF8pq7c61dfeyuPE5Epu5o1bIrunj97HtXVWL7IOwAdlOixF+cccTjuWJ5zd251q4+NlF+DkqjuLz33uJwziUcb3EkNm9xOOcS0XNhB+B2zFsczjnnouItDuecc1HxxOGccy4qnjicc85FxROHc65EEZH9RWSYiLwXdiyllScO51yxEZEXRWSJiMzMd7yziPwiInNF5KbCrqGq81S1T3wjdYUpF3YAzrlS5WXgaeDVvAMiUhYYDJwALASmiMhooCwwKN/jL1bVJcUTqtsRTxzOuWKjquNFJCXf4XbAXFWdByAibwHdVXUQcGoxh+iKwLuqnHNhqwssiLi9MDhWIBGpJSJDgUNF5OZ4B+e25y0O51yJoqrLgb5hx1GaeYvDORe2RUD9iNv1gmMuQXnicM6FbQrQREQaicgewNnA6JBjcoXwxOGcKzYi8iYwCThIRBaKSB9V3QT0Az4FZgPvqOpPYcbpCudFDp1zzkXFWxzOOeei4onDOedcVDxxOOeci4onDuecc1HxxOGccy4qnjicc85FxROHc8VMRMaJSNud3KeHiDSLuH23iBwf/+ic2zlPHM4VkZji+p3pAWxJHKp6u6p+UUzP7VyhPHE4VwgRSQk2GHoVmAncJiJTRORHEbkruE9lEflIRH4QkZkiclZw/DgR+U5EZgQbGFUo4PqrI74/XUReFpF/Ad2Ah0TkexE5IDh+emHXFZFMEblLRKYH5w6O/zvkSiNPHM7tXBPgGeBarNx3O6A1cJiIdAQ6A4tVtZWqtgA+EZGK2KZFZ6nqIVgl6suL8mSqOhGr1TRAVVur6m9554pw3WWq2gYYAvTf5VfsXCE8cTi3c1mqOhk4Mfj6DpgOHIwllRnACSLygIh0UNVs4CDgd1X9NbjGK0DHGMSys+u+H/w7DUiJwfM5tx3fj8O5nVsT/CvAIFV9Nv8dRKQNcDJwr4j8FxhVxGtHFouruFtRmg3Bv5vx328XJ97icK7oPgUuFpEqACJSV0T2FpH9gLWq+jrwENAG+AVIEZHGwWPPB74q4Jp/iUjTYNC9Z8TxHKBqAfcv6nWdixv/ROJcEanqZyLSFJgkIgCrgfOAxthAdi6wEbhcVdeLyEXAuyJSDttzYmgBl70J+BBYCkwFqgTH3wKeF5GrgNMjYijqdZ2LGy+r7pxzLireVeWccy4qnjicc85FxROHc865qHjicM45FxVPHM4556LiicM551xUPHE455yLiicO55xzUfl/LWyuxlrmfRIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "# f = lambda x, y: -(20.0 * y**3 + 9.0 * numpy.pi**2 * (y - y**5)) * numpy.sin(3.0 * numpy.pi * x)\n",
    "# u_true = lambda x, y: (y - y**5) * numpy.sin(3.0 * numpy.pi * x)\n",
    "u_true = lambda x, y: (y**3 - y) * (numpy.cos(2.0 * numpy.pi * x) - 1.0)\n",
    "f = lambda x, y: -4.0 * numpy.pi**2 * (y**3 - y) * numpy.cos(2.0 * numpy.pi * x) + 6.0 * y * (numpy.cos(2.0 * numpy.pi * x) - 1.0)\n",
    "\n",
    "index   = numpy.asarray([2**k for k in numpy.arange(2,7)])\n",
    "delta_x = numpy.zeros(len(index))\n",
    "error   = numpy.zeros(len(index))\n",
    "\n",
    "fig = plt.figure(1)\n",
    "\n",
    "for k,m in enumerate(index):\n",
    "    x, y, U    = solve_9point(m,f)\n",
    "    X, Y       = numpy.meshgrid(x, y)\n",
    "    delta_x[k] = x[1]-x[0]\n",
    "    error[k]   = numpy.linalg.norm(delta_x[k] * (u_true(X,Y) - U), ord=1)\n",
    "    \n",
    "plt.loglog(delta_x,error,'ko')\n",
    "plt.draw()\n",
    "\n",
    "order_C = lambda delta_x, error, order: numpy.exp(numpy.log(error) - order * numpy.log(delta_x))\n",
    "plt.loglog(delta_x, order_C(delta_x[0], error[0], 1.0) * delta_x**1.0, 'r--', label=\"1st Order\")\n",
    "plt.loglog(delta_x, order_C(delta_x[0], error[0], 2.0) * delta_x**2.0, 'b--', label=\"2nd Order\")\n",
    "plt.legend()\n",
    "\n",
    "plt.title('Convergence of Discrete Poisson Problem')\n",
    "plt.xlabel('resolution')\n",
    "plt.ylabel('numerical $L_2$error')\n",
    "plt.show()\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q1-d",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(d)** [15] Show that the 9-point Laplacian can be written as a 5-point Laplacian **plus** a finite difference approximation of $\\frac{1}{6}h^2 u_{xxyy} + \\mathcal O(h^4)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A1-d",
     "locked": false,
     "points": 15,
     "schema_version": 3,
     "solution": true
    }
   },
   "source": [
    "For the 9-point Laplacian we consider the Taylor expansions:  \n",
    "$$\\begin{aligned}\n",
    "    u(x\\pm \\Delta x, y\\pm \\Delta x) &= u(x,y) \\pm \\Delta x ( u_x + u_y ) + \\frac{\\Delta x^2}{2} \\left(u_{xx}+2u_{xy}+u_{yy} \\right) + \\mathcal O(\\Delta x^3)\\\\\n",
    "    u(x\\mp \\Delta x, y\\pm \\Delta x) &= u(x,y) \\mp \\Delta x ( u_x + u_y ) + \\frac{\\Delta x^2}{2} \\left(u_{xx}-2u_{xy}+u_{yy}\\right) + \\mathcal O(\\Delta x^3)\\\\\n",
    "    u(x\\pm \\Delta x, y ) &=  u(x,y) \\pm \\Delta x u_x + \\frac{\\Delta x^2}{2} u_{xx} + \\mathcal O(\\Delta x^3)\\\\\n",
    "    u(x, y\\pm \\Delta x ) &=  u(x,y) \\pm \\Delta x u_y + \\frac{\\Delta x^2}{2} u_{yy} + \\mathcal O(\\Delta x^3)\n",
    "\\end{aligned}$$\n",
    "\n",
    "We take a weighted sum of these equations of the form _once each of the first four and four times each of the last four_.\n",
    "This adds up to:\n",
    "\n",
    "$$\n",
    "  \\frac{1}{6\\Delta x^2} \\left( u(x + \\Delta x, y + \\Delta x) +  u(x - \\Delta x, y + \\Delta x) +  4u(x + \\Delta x, y) +  4u(x, y + \\Delta x) +  \\\n",
    "   u(x - \\Delta x, y - \\Delta x) +  u(x + \\Delta x, y - \\Delta x) +  4u(x - \\Delta x, y) +  4u(x, y - \\Delta x) -  \\\n",
    "   20 u(x,y)\\right)  = \\left(u_{xx}+u_{yy} \\right) + \\mathcal O(\\Delta x^3)\n",
    "$$\n",
    "\n",
    "So we define the 9-point Laplace as\n",
    "$$\n",
    "    \\nabla_9^2 = \\frac{1}{6\\Delta x^2} \\left( u(x + \\Delta x, y + \\Delta x) + u(x - \\Delta x, y + \\Delta x) +  4u(x + \\Delta x, y) + 4 u(x, y + \\Delta x) +  \\\n",
    "   u(x - \\Delta x, y - \\Delta x) +  u(x + \\Delta x, y - \\Delta x) + 4 u(x - \\Delta x, y) + 4 u(x, y - \\Delta x) -  \\\n",
    "   20 u(x,y)\\right) \n",
    "$$\n",
    "\n",
    "\n",
    "For the truncation error, we consider the Taylor expansions:\n",
    "$$\\begin{aligned}\n",
    "   u(x\\pm \\Delta x, y\\pm \\Delta x) &= u(x,y) \\pm \\Delta x ( u_x + u_y ) + \\\n",
    "   \\frac{\\Delta x^2}{2} \\left(u_{xx}+2u_{xy}+u_{yy} \\right) \\pm \\\n",
    "   \\frac{\\Delta x^3}{6} \\left( u_{xxx} + 3 u_{xxy} + 3 u_{xyy} + u_{yyy} \\right) + \\\\\n",
    "   &\\frac{\\Delta x^4}{24} \\left( u_{xxxx} + 4 u_{xxxy} + 6 u_{xxyy} + 4 u_{xyyy} + u_{yyyy} \\right) + \\\n",
    "   \\mathcal O(\\Delta x^5)\\\\\n",
    "    u(x\\mp \\Delta x, y\\pm \\Delta x) &= u(x,y) \\mp \\Delta x ( u_x + u_y ) + \\\n",
    "    \\frac{\\Delta x^2}{2} \\left(u_{xx}-2u_{xy}+u_{yy}\\right) + \\\n",
    "    \\frac{\\Delta x^3}{6} \\left( \\mp u_{xxx} \\pm 3 u_{xxy} \\mp 3 u_{xyy} \\pm u_{yyy} \\right) + \\\\\n",
    "   &\\frac{\\Delta x^4}{24} \\left( u_{xxxx} - 4 u_{xxxy} + 6 u_{xxyy} - 4 u_{xyyy} + u_{yyyy} \\right) + \\\n",
    "   \\mathcal O(\\Delta x^5)\\\\\n",
    "    u(x\\pm \\Delta x, y ) &=  u(x,y) \\pm \\Delta x u_x + \\frac{\\Delta x^2}{2} u_{xx} + \\\n",
    "     \\pm \\frac{\\Delta x^3}{6}u_{xxx} + \\frac{\\Delta x^4}{24} u_{xxxx} +\\mathcal O(\\Delta x^5)\\\\\n",
    "    u(x, y\\pm \\Delta x ) &=  u(x,y) \\pm \\Delta x u_y + \\frac{\\Delta x^2}{2} u_{yy} + \\\n",
    "     \\pm \\frac{\\Delta x^3}{6} u_{yyy} + \\frac{\\Delta x^4}{24} u_{yyyy} + \\mathcal O(\\Delta x^5)\n",
    "\\end{aligned}$$  \n",
    "\n",
    "Plugging those in to the formula for the 9-point Laplacian gives\n",
    "$$\\begin{aligned}\n",
    "    \\nabla_9^2 &= \\Delta u + \\Delta x^2 \\frac{1}{6} \\left( \\frac{12}{24} u_{xxxx}+ \\frac{24}{24} u_{xxyy} + \\frac{12}{24} u_{yyyy}\\right) + \\mathcal O(\\Delta x^5)\\\\\n",
    "               &= \\Delta u + \\Delta x^2 \\frac{1}{12} \\left( u_{xxxx}+ 2 u_{xxyy} + u_{yyyy}\\right) + \\mathcal O(\\Delta x^5),\n",
    "\\end{aligned}$$\n",
    "\n",
    "\n",
    "\n",
    "so that the truncation error is\n",
    "\n",
    "$$\n",
    "    \\tau(\\nabla_9^2) = \\frac{\\Delta x^2}{12} \\left( u_{xxxx}+ 2 u_{xxyy} + u_{yyyy}\\right) + \\mathcal O(\\Delta x^5)\n",
    "$$\n",
    "\n",
    "So, the error is still of order $\\mathcal O(\\Delta x^2)$.\n",
    "\n",
    "Let us assume that $\\Delta x = \\Delta y$. The 9-point Laplacian is given by \n",
    "\n",
    "$$\n",
    "    \\nabla_9^2 = \\frac{1}{6\\Delta x^2} \\left( u(x + \\Delta x, y + \\Delta x) + u(x - \\Delta x, y + \\Delta x) +  4u(x + \\Delta x, y) + 4 u(x, y + \\Delta x) +  \\\n",
    "   u(x - \\Delta x, y - \\Delta x) +  u(x + \\Delta x, y - \\Delta x) + 4 u(x - \\Delta x, y) + 4 u(x, y - \\Delta x) -  \\\n",
    "   20 u(x,y)\\right) \n",
    "$$\n",
    "\n",
    "From the lecture we know that the truncation error for the 5-point Laplacian is\n",
    "\n",
    "$$\n",
    "    \\tau(\\nabla_5^2) = \\frac{\\Delta x^2}{12} \\left( u_{xxxx} + u_{yyyy} \\right) + \\mathcal O(\\Delta x^4)\n",
    "$$\n",
    "\n",
    "Subtracting this result from the answer above, we obtain\n",
    "\n",
    "$$\n",
    "    \\tau(\\nabla_9^2 - \\nabla_5^2) = \\Delta x^2 \\frac{1}{6} \\left( u_{xxyy} \\right) + \\mathcal O(\\Delta x^4)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q1-e",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(e)** [10] Modify your function to use the trick introduced in class that will cause the 9-point Laplacian stencil to become 4th order accurate.  Show that this is true via a convergence study."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A1-e",
     "locked": false,
     "points": 10,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "import scipy.sparse as sparse\n",
    "import scipy.sparse.linalg as linalg\n",
    "\n",
    "def solve_9point_plus(m, f, laplace_f):\n",
    "    x = numpy.linspace(0, 1.0, m + 2)\n",
    "    y = numpy.linspace(0, 1.0, m + 2)\n",
    "    delta_x = 1.0 / (m + 1)\n",
    "\n",
    "    # Construct A\n",
    "    e = numpy.ones(m)\n",
    "    T = sparse.spdiags([4.0 * e, -20.0 * e, 4.0 * e], [-1, 0, 1], m, m)\n",
    "    I = sparse.eye(m)\n",
    "    S = sparse.spdiags([e, e], [-1, 1], m, m)\n",
    "    OI = sparse.spdiags([e, 4.0 * e, e], [-1, 0, 1], m, m)\n",
    "    A = (sparse.kron(I, T) + sparse.kron(S, OI)).tocsr()\n",
    "    A /= 6.0 * delta_x**2\n",
    "    \n",
    "    # Right-hand-side\n",
    "    X, Y = numpy.meshgrid(x[1:-1], y[1:-1])\n",
    "    b = (f(X, Y) + delta_x**2 / 12.0 * laplace_f(X, Y)).reshape(m**2, order='F')\n",
    "\n",
    "    # Solve\n",
    "    U = numpy.zeros((m+2, m+2))\n",
    "    U[1:-1, 1:-1] = linalg.spsolve(A, b).reshape((m, m), order='F')\n",
    "\n",
    "    return x, y, U\n",
    "\n",
    "\n",
    "u_true = lambda x, y: (y**3 - y) * (numpy.cos(2.0 * numpy.pi * x) - 1.0)\n",
    "f = lambda x, y: -4.0 * numpy.pi**2 * (y**3 - y) * numpy.cos(2.0 * numpy.pi * x) + 6.0 * y * (numpy.cos(2.0 * numpy.pi * x) - 1.0)\n",
    "laplace_f = lambda x, y: 16.0 * numpy.pi**4 * (y**3 - y) * numpy.cos(2.0 * numpy.pi * x) - 48.0 * numpy.pi**2 * y * numpy.cos(2.0 * numpy.pi * x)\n",
    "\n",
    "index   = numpy.asarray([2**k for k in numpy.arange(2,7)])\n",
    "delta_x = numpy.zeros(len(index))\n",
    "error   = numpy.zeros(len(index))\n",
    "\n",
    "fig = plt.figure(1)\n",
    "\n",
    "for k,m in enumerate(index):\n",
    "    x, y, U    = solve_9point_plus(m, f, laplace_f)\n",
    "    X, Y       = numpy.meshgrid(x, y)\n",
    "    delta_x[k] = x[1]-x[0]\n",
    "    error[k]   = numpy.linalg.norm(delta_x[k] * (u_true(X,Y) - U), ord=1)\n",
    "    \n",
    "plt.loglog(delta_x,error,'ko')\n",
    "plt.draw()\n",
    "\n",
    "order_C = lambda delta_x, error, order: numpy.exp(numpy.log(error) - order * numpy.log(delta_x))\n",
    "plt.loglog(delta_x, order_C(delta_x[0], error[0], 2.0) * delta_x**2.0, 'b--', label=\"2nd Order\")\n",
    "plt.loglog(delta_x, order_C(delta_x[0], error[0], 4.0) * delta_x**4.0, 'g--', label=\"4th Order\")\n",
    "plt.legend()\n",
    "\n",
    "plt.title('Convergence of Discrete Poisson Problem')\n",
    "plt.xlabel('resolution')\n",
    "plt.ylabel('numerical $L_2$error')\n",
    "plt.show()\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q2",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "## Question 2\n",
    "\n",
    "Let us consider some modifications to the Gauss-Seidel method we discussed and when one version might be better than the other."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q2-a",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(a)** [5] The Gauss-Seidel method for the discretization of $u''(x) = f(x)$\n",
    "takes the form \n",
    "$$\n",
    "    U_i^{(k+1)} = \\frac{1}{2} (U_{i+1}^{(k)} + U_{i-1}^{(k+1)}) - f(x_i) \\frac{\\Delta x^2}{2}\n",
    "$$\n",
    "if we assume we are marching forwards across the grid,\n",
    "for $i=1,~2,~\\ldots,~m$.  We can also define a *backwards Gauss-Seidel\n",
    "method* by setting\n",
    "$$\n",
    "    U_i^{(k+1)} = \\frac{1}{2} (U_{i+1}^{(k+1)} + U_{i-1}^{(k)}) - f(x_i) \\frac{\\Delta x^2}{2}\n",
    "$$\n",
    "Show that this is a matrix splitting method of the type described in the lecture notes and what the splitting is."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A2-a",
     "locked": false,
     "points": 5,
     "schema_version": 3,
     "solution": true
    }
   },
   "source": [
    "We just need to simply show that we can write the iterative scheme with appropriate $M$ and $N$.  Using the terminology in the lecture notes we have $M = D-U$ and $N=L$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q2-b",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(b)** [5] Implement the backwards Gauss-Seidel method and show that the convergence is at the same rate as the forward version (do this computationally, you do not need to do this analytically).  Use the same expected iteration count as with the forwards method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "A2-b",
     "locked": false,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [],
   "source": [
    "def solve_BGS(a, b, alpha, beta, m, f):\n",
    "    ### BEGIN SOLUTION\n",
    "    \n",
    "    # Descretization\n",
    "    x_bc = numpy.linspace(a, b, m + 2)\n",
    "    x = x_bc[1:-1]\n",
    "    delta_x = (b - a) / (m + 1)\n",
    "\n",
    "    # Expected iterations needed\n",
    "    iterations_GS = int(2.0 * numpy.log(delta_x) / numpy.log(1.0 - numpy.pi**2 * delta_x**2))\n",
    "\n",
    "    # Solve system\n",
    "    # Initial guess for iterations\n",
    "    U = numpy.zeros(m + 2)\n",
    "    U[0] = alpha\n",
    "    U[-1] = beta\n",
    "    success = False\n",
    "    for k in range(iterations_GS):\n",
    "        for i in range(m, 0, -1):\n",
    "            U[i] = 0.5 * (U[i+1] + U[i-1]) - f(x_bc[i]) * delta_x**2 / 2.0\n",
    "    ### END SOLUTION\n",
    "\n",
    "    return x_bc, U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "T2-b",
     "locked": true,
     "points": 5,
     "schema_version": 3,
     "solution": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.595681653803634e-05\n",
      "Success!\n"
     ]
    }
   ],
   "source": [
    "# Problem setup\n",
    "a = 0.0\n",
    "b = 1.0\n",
    "alpha = 0.0\n",
    "beta = 3.0\n",
    "f = lambda x: numpy.exp(x)\n",
    "u_true = lambda x: (4.0 - numpy.exp(1.0)) * x - 1.0 + numpy.exp(x)\n",
    "\n",
    "x, U = solve_BGS(a, b, alpha, beta, 150, f)\n",
    "error = numpy.linalg.norm((x[1] - x[0]) * (u_true(x) - U), ord=1)\n",
    "print(error)\n",
    "assert(error < 1e-4)\n",
    "### BEGIN HIDDEN TESTS\n",
    "print(\"Success!\")\n",
    "### END HIDDEN TESTS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q2-c",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(c)** [10] Modify the code so that it solves the steady-state problem\n",
    "$$\n",
    "    \\epsilon u''(x) - u'(x) = f(x)\n",
    "$$\n",
    "we had before with the same boundaries and $f(x)$.  This time however use the one-sided approximation to the first derivative\n",
    "$$\n",
    "    \\frac{U_i - U_{i-1}}{\\Delta x}.\n",
    "$$\n",
    "Test both forward and backward Gauss-Seidel for the resulting linear system.  Use the value $\\epsilon = 0.01$, $m = 100$, and a maximum number of iterations of $250$.  Plot the convergence of both approaches."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A2-c1",
     "locked": false,
     "points": 10,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = 0.0\n",
    "b = 1.0\n",
    "alpha = 1.0\n",
    "beta = 3.0\n",
    "epsilon = 0.01\n",
    "u_true = lambda x: alpha + x + (beta - alpha - 1.0) * (numpy.exp(x / epsilon) - 1.0) / (numpy.exp(1.0 / epsilon) - 1.0)\n",
    "\n",
    "### BEGIN SOLUTION\n",
    "# Descretization\n",
    "m = 100\n",
    "x_bc = numpy.linspace(a, b, m + 2)\n",
    "x = x_bc[1:-1]\n",
    "delta_x = (b - a) / (m + 1)\n",
    "\n",
    "# max_iterations = int(2.0 * numpy.log(delta_x) / numpy.log(1.0 - numpy.pi**2 * delta_x**2))\n",
    "max_iterations = 250\n",
    "convergence = numpy.zeros((2, max_iterations))\n",
    "\n",
    "# Forward GS\n",
    "U_f = numpy.zeros(m + 2)\n",
    "U_f[0] = alpha\n",
    "U_f[-1] = beta\n",
    "for k in range(max_iterations):\n",
    "    for i in range(1, m + 1):\n",
    "        U_f[i] = (epsilon * U_f[i+1] + (epsilon + delta_x) * U_f[i-1] + delta_x**2) / (2.0 * epsilon + delta_x)\n",
    "    convergence[0, k] = numpy.linalg.norm(delta_x * (u_true(x_bc) - U_f), ord=1)\n",
    "\n",
    "# Backward GS\n",
    "U_b = numpy.zeros(m + 2)\n",
    "U_b[0] = alpha\n",
    "U_b[-1] = beta\n",
    "for k in range(max_iterations):\n",
    "    for i in range(m, 0, -1):\n",
    "        U_b[i] = (epsilon * U_b[i+1] + (epsilon + delta_x) * U_b[i-1] + delta_x**2) / (2.0 * epsilon + delta_x)\n",
    "    convergence[1, k] = numpy.linalg.norm(delta_x * (u_true(x_bc) - U_b), ord=1)    \n",
    "    \n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(x_bc, U_f, 'r+', label=\"Forward GS\")\n",
    "axes.plot(x_bc, U_b, 'bx', label=\"Backward GS\")\n",
    "axes.plot(x_bc, u_true(x_bc), 'k-', label=\"True Solution\")\n",
    "axes.set_title(\"Solutions\")\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$u(x)$\")\n",
    "axes.legend(loc=2)\n",
    "\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.semilogy(convergence[0, :], 'r', label=\"Forward\")\n",
    "axes.semilogy(convergence[1, :], 'b', label=\"Backward\")\n",
    "axes.set_title(\"Convergence Rates\")\n",
    "axes.set_xlabel(\"k\")\n",
    "axes.set_ylabel(\"$ ||\\Delta x(u - U)||_1$\")\n",
    "axes.legend(loc=1)\n",
    "\n",
    "plt.show()\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q2-d",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(d)** [5] Explain intuitively why sweeping in one direction works so much better than\n",
    "in the other.\n",
    "\n",
    "**Hint**: Note that this equation is the steady equation for an advection-diffusion PDE \n",
    "$$\n",
    "    u_t(x,t) + u_x(x,t) = \\epsilon u_{xx}(x,t) - f(x).\n",
    "$$  \n",
    "You might consider how the methods behave in the case $\\epsilon = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "grade_id": "A2-d",
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   "source": [
    "There should be marked difference between the forward and backward sweeps due to the one-sided approximation.  Clearly the non-updated sweep in the backward direction is impacting the convergence.  This has more to do with the upwinding of the advective term than anything else.  If $\\epsilon$ is small than this acts more like a hyperbolic PDE in which we would expect there to be more of instability one direction.  Given that we have not covered this, the above explanation is fine, the sweep direction clearly has a preference when using one-sided differences due to the dependence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "grade": false,
     "grade_id": "Q3",
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     "schema_version": 3,
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    }
   },
   "source": [
    "## Question 3 - Poisson on a Rectangle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q3-a",
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    }
   },
   "source": [
    "**(a)** [15] Write a function that solves the Poisson problem \n",
    "$$\n",
    "    \\nabla^2 u = \\frac{5}{4} e^{x + y/2}\n",
    "$$\n",
    "on $\\Omega = [0, 1] \\times [0, 2]$ using a five point stencil.  Use the true solution of the problem\n",
    "$$\n",
    "    u(x, y) = e^{x + y / 2}\n",
    "$$\n",
    "to set the boundary conditions.  Allow for $\\Delta x$ and $\\Delta y$ to be non-uniform."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "A3-a",
     "locked": false,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [],
   "source": [
    "import scipy.sparse as sparse\n",
    "import scipy.sparse.linalg as linalg\n",
    "\n",
    "def solve_poisson(X, Y, f):\n",
    "    # BEGIN SOLUTION\n",
    "    \n",
    "    # Grid parameters\n",
    "    delta_x = X[1, 0] - X[0, 0]\n",
    "    delta_y = Y[0, 1] - Y[0, 0]\n",
    "    \n",
    "    # Construct A\n",
    "    m = X.shape[0] - 2\n",
    "    n = Y.shape[1] - 2\n",
    "    \n",
    "    e = numpy.ones(m)\n",
    "    en = numpy.ones(n)\n",
    "    \n",
    "    T = sparse.spdiags([e / delta_x**2, -2.0 * e / delta_x**2 - 2.0 * e / delta_y**2, e / delta_x**2], [-1, 0, 1], m, m)\n",
    "    S = sparse.spdiags([e / delta_y**2], [0], m, m)\n",
    "    \n",
    "    I = sparse.eye(n)\n",
    "    I2 = sparse.spdiags([en], [1], n, n)\n",
    "    I3 = sparse.spdiags([en], [-1], n, n)\n",
    "    \n",
    "    A = sparse.kron(I, T) + sparse.kron(I2, S) + sparse.kron(I3, S)\n",
    "    \n",
    "    # Construct rhs\n",
    "    b = f(X[1:-1, 1:-1], Y[1:-1, 1:-1])\n",
    "\n",
    "    # Boundaries - Note that this looks weird due to the resulting ordering\n",
    "    b[0, :] -= numpy.exp(X[0, 1:-1] + Y[0, 1:-1] / 2.0) / delta_x**2    \n",
    "    b[-1, :] -= numpy.exp(X[-1, 1:-1] + Y[-1, 1:-1] / 2.0) / delta_x**2\n",
    "    b[:, 0] -= numpy.exp(X[1:-1, 0] + Y[1:-1, 0] / 2.0) / delta_y**2\n",
    "    b[:, -1] -= numpy.exp(X[1:-1, -1] + Y[1:-1, -1] / 2.0) / delta_y**2\n",
    "\n",
    "    U = numpy.exp(X + Y / 2.0)\n",
    "    U[1:-1, 1:-1] = linalg.spsolve(A, b.reshape(m*n, order='F')).reshape((m, n), order='F')\n",
    "    \n",
    "    # END SOLUTION\n",
    "    return U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "T3-a",
     "locked": true,
     "points": 15,
     "schema_version": 3,
     "solution": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success!\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/codio/.local/lib/python3.6/site-packages/ipykernel_launcher.py:25: MatplotlibDeprecationWarning: shading='flat' when X and Y have the same dimensions as C is deprecated since 3.3.  Either specify the corners of the quadrilaterals with X and Y, or pass shading='auto', 'nearest' or 'gouraud', or set rcParams['pcolor.shading'].  This will become an error two minor releases later.\n",
      "/home/codio/.local/lib/python3.6/site-packages/ipykernel_launcher.py:31: MatplotlibDeprecationWarning: shading='flat' when X and Y have the same dimensions as C is deprecated since 3.3.  Either specify the corners of the quadrilaterals with X and Y, or pass shading='auto', 'nearest' or 'gouraud', or set rcParams['pcolor.shading'].  This will become an error two minor releases later.\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Grid including boundaries\n",
    "# Note that the x and y arrays include the boundary location\n",
    "m = 15\n",
    "n = 25\n",
    "delta_x = 1.0 / (m + 1)\n",
    "delta_y = 2.0 / (n + 1)\n",
    "x = numpy.linspace(0.0, 1.0, m + 2)\n",
    "y = numpy.linspace(0.0, 2.0, n + 2)\n",
    "X, Y = numpy.meshgrid(x, y)\n",
    "# Transpose these so that the coordinates match up to (i, j)\n",
    "X = X.transpose()\n",
    "Y = Y.transpose()\n",
    "\n",
    "f = lambda x, y: 1.25 * numpy.exp(x + y / 2.0)\n",
    "u_true = lambda x, y: numpy.exp(x + y / 2.0)\n",
    "U = solve_poisson(X, Y, f)\n",
    "assert numpy.linalg.norm(max(delta_x, delta_y) * (u_true(X, Y) - U), ord=1) < 1e-3\n",
    "### BEGIN HIDDEN TESTS\n",
    "print(\"Success!\")\n",
    "### END HIDDEN TESTS\n",
    "\n",
    "fig = plt.figure()\n",
    "fig.set_figwidth(fig.get_figwidth())\n",
    "axes = fig.add_subplot(1, 2, 1, aspect='equal')\n",
    "plot = axes.pcolor(X, Y, U, vmax=7.0, vmin=0.0, cmap=plt.get_cmap(\"Blues\"))\n",
    "fig.colorbar(plot, label=\"$U$\")\n",
    "axes.set_title(\"Computed Solution\")\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"y\")\n",
    "axes = fig.add_subplot(1, 2, 2, aspect='equal')\n",
    "plot = axes.pcolor(X, Y, u_true(X, Y), vmax=7.0, vmin=0.0, cmap=plt.get_cmap(\"Blues\"))\n",
    "fig.colorbar(plot, label=\"$u(x,t)$\")\n",
    "axes.set_title(\"True Solution\")\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"y\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": false,
     "grade_id": "Q3-b",
     "locked": true,
     "schema_version": 3,
     "solution": false
    }
   },
   "source": [
    "**(b)** [15] Keeping $\\Delta x$ constant study the convergence behavior as $\\Delta y \\rightarrow 0$.  Do the same for $\\Delta x$.  Comment on what you observe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A3-b1",
     "locked": false,
     "points": 10,
     "schema_version": 3,
     "solution": true
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### BEGIN SOLUTION\n",
    "f = lambda x, y: 1.25 * numpy.exp(x + y / 2.0)\n",
    "u_true = lambda x, y: numpy.exp(x + y / 2.0)\n",
    "\n",
    "# Perform x-sweep\n",
    "n = 64\n",
    "delta_y = 2.0 / (n + 1)\n",
    "x_norm = numpy.empty(5)\n",
    "for (i, m) in enumerate([2**k for k in range(4, 9)]):\n",
    "    delta_x = 1.0 / (m + 1)\n",
    "    x = numpy.linspace(0.0, 1.0, m + 2)\n",
    "    y = numpy.linspace(0.0, 2.0, n + 2)\n",
    "    X, Y = numpy.meshgrid(x, y)\n",
    "    X = X.transpose()\n",
    "    Y = Y.transpose()\n",
    "\n",
    "    U = solve_poisson(X, Y, f)\n",
    "    x_norm[i] = numpy.linalg.norm(max(delta_x, delta_y) * (u_true(X, Y) - U), ord=1)\n",
    "\n",
    "# Perform y-sweep\n",
    "m = 64\n",
    "delta_x = 1.0 / (m + 1)\n",
    "y_norm = numpy.empty(5)\n",
    "for (i, n) in enumerate([2**k for k in range(4, 9)]):\n",
    "    delta_y = 2.0 / (n + 1)\n",
    "    x = numpy.linspace(0.0, 1.0, m + 2)\n",
    "    y = numpy.linspace(0.0, 2.0, n + 2)\n",
    "    X, Y = numpy.meshgrid(x, y)\n",
    "    X = X.transpose()\n",
    "    Y = Y.transpose()\n",
    "\n",
    "    U = solve_poisson(X, Y, f)\n",
    "    y_norm[i] = numpy.linalg.norm(max(delta_x, delta_y) * (u_true(X, Y) - U), ord=1)\n",
    "    \n",
    "# Plots\n",
    "fig = plt.figure()\n",
    "fig.set_figwidth(fig.get_figwidth() * 2)\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "axes.semilogy([2**k for k in range(4, 9)], x_norm)\n",
    "axes.set_title(\"$\\Delta x$ Convergence\")\n",
    "axes.set_xlabel(\"m\")\n",
    "axes.set_ylabel(\"$\\ell_1$ grid norm\")\n",
    "\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "axes.semilogy([2**k for k in range(4, 9)], y_norm)\n",
    "axes.set_title(\"$\\Delta y$ Convergence\")\n",
    "axes.set_xlabel(\"n\")\n",
    "axes.set_ylabel(\"$\\ell_1$ grid norm\")\n",
    "\n",
    "plt.show()\n",
    "\n",
    "### END SOLUTION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbgrader": {
     "grade": true,
     "grade_id": "A3-b2",
     "locked": false,
     "points": 5,
     "schema_version": 3,
     "solution": true
    }
   },
   "source": [
    "One should see that eventually the convergence does not extend past a certain point due to the limited accuracy in the other direction."
   ]
  }
 ],
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