{ "cells": [ { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from __future__ import division \n", "from IPython.display import Latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# On Galerkin approximations for the QG equations\n", "## Supplementary material \n", "

\n", "### Cesar B. Rocha*, William R. Young, and Ian Grooms \n", "

\n", "### Winter 2015\n", "

\n", "*Scripps Institution of Oceanography, University of California, San Diego, 9500 Gilman Dr. MC 0213, La Jolla, CA/USA, " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Can we get a sine out of cosines? \n", "##An elementary example of Galerkin approximation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the linear boundary value problem\n", "\n", "\\begin{equation}\n", "\\phi'' + \\phi = 0\\, ,\n", "\\end{equation}\n", "with boundary conditions\n", "\n", "\\begin{equation}\n", "@z = 0 \\qquad \\phi' = 1 \\, ,\n", "\\end{equation}\n", "and\n", "\\begin{equation}\n", "@z = \\pi \\qquad \\phi' = -1 \\, .\n", "\\end{equation}\n", "

\n", "The most general exact solution is\n", "\\begin{equation}\n", "\\phi(z) = \\sin z + C\\cos z\\, ,\n", "\\end{equation}\n", "

\n", "\n", "where $C$ is an undetermined constant. A solution can be found as a linear combination of the eigenfunctions of the linear operator $D_2 \\equiv d\\,^2/d\\, z^2$:\n", "\n", "\\begin{equation}\n", "\\phi = \\sum_{n=0}^{\\infty} \\phi_n\\,\\mathrm{e}_n\\,.\n", "\\end{equation}\n", "\n", "If we define the $\\mathrm{e}_n$ as the eigenfunctions of $D_2$ with inhomogeneous boundary conditions, the series above is truncated and gives the exact solution. With homogeneous boundary conditions the series is infinite. We define\n", "\n", "\\begin{equation}\n", "D_2 \\, \\mathrm{e}_n = -\\lambda_n^2\\, \\mathrm{e}_n\\,,\n", "\\end{equation}\n", "and\n", "\n", "\\begin{equation}\n", "@z = 0\\,,\\pi:\\qquad \\mathrm{e}_n' = 0\\,.\n", "\\end{equation}\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " The Galerkin approximation represents the solution as a truncated series in this orthogonal basis:\n", "\\begin{equation}\n", "\\phi^{\\mathrm{N}}_G = \\sum_{n=0}^{\\mathrm{N}} \\phi_n\\,\\mathrm{e}_n\\,,\n", "\\end{equation}\n", "

\n", "The eigenproblem above is a simplified Sturm-Liouville problem (e.g. Strang CSE book). The eigenfunctions are orthogonal because the operator $D_2$ is self-ajoint. It is convenient to normalize the eigenfunctions $\\mathrm{e}_n$ to have unit $L_2$-norm\n", "\n", "\\begin{equation}\n", "\\frac{1}{\\pi}\\int_{0}^{\\pi} \\mathrm{e}_n\\,\\mathrm{e}_m \\,d z= \\delta_{mn} \\,,\n", "\\end{equation}\n", "

\n", "Thus we have \n", "\n", "\\begin{equation}\n", "\\mathrm{e}_0 = 1 \\qquad \\text{and} \\qquad \\mathrm{e}_n = \\sqrt{2} \\cos n \\,z\\,.\n", "\\end{equation}\n", "\n", "The coefficients $\\phi_n$ are determined to minimize the $L_2$-norm of the residual \n", "\n", "\\begin{equation}\n", "E(\\phi_0, \\phi_1, \\cdots , \\phi_N) \\equiv \\frac{1}{\\pi} \\int_{0}^{\\pi} \\left( \\phi - \\sum_{n=0}^{\\mathrm{N}} \\phi_n \\mathrm{e}_n\\right)^2 \\, d\\, z \\, .\n", "\\end{equation}\n", "Thus\n", "\n", "\n", "\\begin{equation}\n", "\\phi_n = \\frac {1}{\\pi} \\int_{0}^{\\pi} \\mathrm{e}_n \\phi \\, d z\\, .\n", "\\end{equation}\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That is, the n'th coefficient is the projection of the true solution onto the n'th eigenfunction. **The Galerkin approximation is therefore the best least-squares approximation using the truncated series in the basis spanned by $\\mathrm{e}_n$, $n=0,\\,1,\\,2,\\ldots,\\,\\mathrm{N}$**. \n", "

\n", "At this point, one might be tempted to introduce the Galerkin series $\\phi^{\\mathrm{N}}_G$ into the governing equation to obtain $\\phi_n$. This **naïve approach** leads to\n", "\n", "\\begin{equation}\n", "\\sum_{n=0}^{\\mathrm{N}} (1-n^2)\\phi_n\\,\\mathrm{e}_n = 0\\,,\n", "\\end{equation}\n", "

\n", "from which one might wrongly conclude that $\\phi_n = 0,\\,n\\neq 1$; $\\phi_1$ is undetermined (this is correct). This approach only capturues part of the exact solution $C\\,\\cos z$; it does not approximates the sine term by a seires of cosines. This **naïve approach** gives the wrong answer.\n", "\n", "

\n", "Instead, to determine $\\phi_n$ we project the exact equation onto the n'th eigenfunction $\\mathrm{e}_n$. That is, we multiply the exact equation by $\\tfrac{1}{\\pi}\\mathrm{e}_n$ and integrate over the domain:\n", "

\n", "\n", "\\begin{equation}\n", "\\frac{1}{\\pi}\\int_0^{\\pi}\\mathrm{e}_n\\,y'' \\,dz + \\frac{1}{\\pi}\\int_0^{\\pi}\\mathrm{e}_n\\,y \\,dz = 0\\,.\n", "\\end{equation}\n", "

\n", "We have to move the derivatives off $y$. As most of exciting things in applied mathematics, this is possible using integration by parts\n", "\n", "\\begin{equation}\n", "\\frac{1}{\\pi}\\int_0^{\\pi}\\!\\!\\!\\!\\mathrm{e}_n\\,y'' \\,dz = \\frac{1}{\\pi}\\int_0^{\\pi}\\!\\!\\!\\!\\mathrm{e}_n''\\,y \\,\\,dz + \\frac{1}{\\pi}\\Big[\\mathrm{e}_n\\,y' - \\mathrm{e}_n'\\,y \\Big]_0^{\\pi}\\,.\n", "\\end{equation}\n", "

\n", "Using the definition of $\\mathrm{e}_n$ and the boundary conditions on $y$ and $\\mathrm{e}_n$, we finally obtain\n", "

\n", "\\begin{equation}\n", "\\phi_0 = \\frac{2}{\\pi}\\qquad \\text{and}\\qquad\\phi_n = \\frac{2\\,\\sqrt{2}}{\\pi\\left(1-n^2\\right)}\\,,\\qquad n \\text{ even}\\,\\,.\n", "\\end{equation}\n", "

\n", "Notice that $\\phi_1$ (the coefficient of $\\cos z$) is undetermined as in the exact solution. Thus, the Galerkin approximate solution is \n", "\n", "\\begin{equation}\n", "\\phi^{\\mathrm{N}}_G = \\frac{2}{\\pi} + \\phi_1 \\cos z+ \\frac{4}{\\pi} \\sum_{n=2,\\text{ even}}^{\\mathrm{N}} \\frac{\\cos n z}{1-n^2}\\,.\n", "\\end{equation}\n", "

\n", "With $\\phi_1=0$, **the Galerkin series $\\phi^{\\mathrm{N}}_G$ is the best least-squares approximation for $\\sin z$, $z \\in [0,\\pi]$**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Comparison with exact solution\n", "We now comparare the Galerkin approximation againt the exact solution. **For simplicity we take $C = \\phi_1 = 0$.**" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np\n", "from numpy import sqrt,pi,sin,cos" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define the domain and compute the exact solution" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": true }, "outputs": [], "source": [ "z = np.linspace(0,pi,100)\n", "phi_exact = sin(z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A simple function to evaluate the Galerkin series" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def Galerkin_solution(N,z):\n", " \"\"\"Computes the Galerkin solution with phi_1 = 0 \n", " -------------------------------\n", " Inputs: N: truncation index \n", " z: array of depths \n", " ------------------------------- \"\"\"\n", " \n", " n = np.arange(2,N+1,2)\n", " phi = np.zeros(z.size)\n", " for i in range(z.size):\n", " phi[i] = 2/pi + (4/pi)*( ( cos(n*z[i])/(1-n**2) ).sum() )\n", " \n", " return phi" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now compute the galerkin solutin for various $\\mathrm{N}$" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [], "source": [ "N = np.arange(0,20,2)\n", "\n", "phi_galerkin = np.zeros((N.size,z.size))\n", "\n", "for i in range(N.size):\n", " phi_galerkin[i,:] = Galerkin_solution(N[i],z)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "plt.rcParams.update({'font.size': 25, 'legend.handlelength' : 1.25})\n", "\n", "import seaborn as sns\n", "sns.set(style=\"darkgrid\")\n", "sns.set_context(\"paper\", font_scale=3., rc={\"lines.linewidth\": 1.5})\n", "sns.set_style(\"darkgrid\",{'grid_color':.95})" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAqAAAAKMCAYAAAA0UgqIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xe8JFd95/1PVYfqnPumvnnCnZw0SghkEGAEBtvYYMCY\n", "9MK7i+M+3ue1z2txYIVNWNZmsb1gnDCY8BgvGPBjMFk5S6M0mnDD3Bz69u2cu7qr6vnjSoOENNKM\n", "5k7dCb/3X5qe7upTP9Wc+73n1DmlWJaFEEIIIYQQdlE3uwFCCCGEEOLKIgFUCCGEEELYSgKoEEII\n", "IYSwlQRQIYQQQghhKwmgQgghhBDCVhJAhRBCCCGErSSACiGEEEIIW0kAFUIIIYQQtpIAKoQQQggh\n", "bCUBVAghhBBC2EoCqBBCCCGEsJUEUCGEEEIIYSsJoEIIIYQQwlbOzW7ApWJsbOx64G+APWfx9v8+\n", 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"display_data" } ], "source": [ "plt.figure(figsize=(10,10))\n", "plt.plot(phi_exact,z,'k',linewidth=3.,alpha=.8,label='exact')\n", "for i in range(N.size):\n", " plt.plot(phi_galerkin[i,:],z,linewidth=2,alpha=.5,label=str(N[i]))\n", "plt.xlabel(r'$\\phi,\\,\\phi^{\\mathrm{N}}_G$')\n", "plt.ylabel(r'z')\n", "plt.yticks([0,pi/4,pi/2,3*pi/4,pi],[r'$0$',r'$\\pi/4$',r'$\\pi/2$',r'$3\\pi/4$',r'$\\pi$'])\n", "plt.ylim(0,pi)\n", "plt.legend(loc=(.04,.15), title = r'$\\mathrm{N} $')\n", "plt.savefig('galerk_sin_cos',format='eps')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "While the galerkin approximation $\\phi^{\\mathrm{N}}_G$ does not satisfy the inhomogeneous boundary conditions the solution with moderate $\\mathrm{N}$ provides an acceptable approximation to the exact solution $\\phi$. However, the convergence near the boundaries is slow. To see this, note that the error at the boundaries is\n", "\n", "\\begin{equation}\n", "E_b = \\frac{4}{\\pi}\\sum_{n=N+1, \\text{even}}^{\\infty}\\frac{1}{1-n^{2}}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence\n", "\n", "\\begin{equation}\n", "E_b \\sim\\! -\\frac{4}{\\pi}\\sum_{n=N+1, \\text{even}}^{\\infty}\\frac{1}{n^{2}}\\approx -\\frac{4}{\\pi}\\int \\frac{1}{\\mathrm{N}^2} \\, dz= \\frac{4}{\\pi \\mathrm{N}}\n", "\\end{equation}\n", "\n", "

\n", "Thus the Galerkin series converges at rate $\\sim\\!\\mathrm{N}^{-1}$ at the boundaries. Let's test this against our computational estimates." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [], "source": [ "N = np.arange(0,1000,2)\n", "phi_galerkin2 = np.zeros((N.size,z.size))\n", "\n", "for i in range(N.size):\n", " phi_galerkin2[i,:] = Galerkin_solution(N[i],z)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that the absolute error at $z=0$ (or at $z=\\phi$) is simply $\\phi^{\\mathrm{N}}_G (0)$. We plot this value against $\\mathrm{N}$ in a log$_{10}\\times\\log_{10}$ space. For reference we also plot the line $\\mathrm{N}^{-1}$. " ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAwwAAAIaCAYAAACNh8pJAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlcVfed//HXvYCggCIKsgmoiUeN+75kMcbEaNTGNF2m\n", "6bTpOmk7baeTmSadmSZt07RN0s50Mm3STJvJbzJZW5u0RhOzqFkU9w1w+bqyCgoqiguy3PP744IB\n", "4SogcM6B9/Px6OOa+z1cPmn7xe+bcz7fr8+2bURERERERFrid7oAERERERFxLwUGEREREREJSYFB\n", "RERERERCUmAQEREREZGQFBhERERERCSkcKcLcDPLsm4GfgaMAU4AzwE/McbUOVqYiIiIiEgX0R2G\n", "ECzLmga8BeQDdwG/BR4EfulkXSIiIiIiXcmncxhaZlnWSiDBGDOp0Xv/ADwBpBtjShwrTkRERESk\n", "i+gOQwssy4oEZgOvXTL0JyAMuL2raxIRERERcYICQ8uGAr0A0/hNY0wxcB4Y4URRIiIiIiJdTYGh\n", "Zf3qX0+3MFYJ9O3CWkREREREHNMtdkmyLMsPrAeGGmMSLnNdf+BhYAmQDJQRbGz+iTGmoNGlVwpS\n", "avwQERERkR6hu9xh+Ckwhcss5OvDQhbwHYJ3EHYAkcCXgR2WZY1pdHlF/WtsCx8VC5zqgJpFRERE\n", "RFzP04HBsiyfZVk/Irjd6ZX8HrCAFUCqMWYqkAr8PyAOeKX+TgXAQaAOuOaS75cK9Ab2dET9IiIi\n", 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