05_elliptic.ipynb

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    "<table>\n",
    " <tr align=left><td><img align=left src=\"https://i.creativecommons.org/l/by/4.0/88x31.png\">\n",
    " <td>Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Kyle T. Mandli</td>\n",
    "</table>"
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   "source": [
    "%matplotlib inline\n",
    "import numpy\n",
    "import matplotlib.pyplot as plt"
   ]
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   "source": [
    "# Elliptic Equations"
   ]
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   "source": [
    "Higher dimensional elliptic equations in general have the form\n",
    "$$\n",
    "    a_1 u_{xx} + a_2 u_{xy} + a_3 u_{yy} + a_4 u_x + a_5 u_y + a_6 u = f\n",
    "$$\n",
    "where the coefficients must satisfy\n",
    "$$\n",
    "    a_2^2 - 4 a_1 a_3 < 0.\n",
    "$$"
   ]
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    "One natural place again that equations like this come up is again in the steady-state problems arising from the heat equation.  In two dimensions we have\n",
    "$$\n",
    "    u_t = (\\kappa u_x)_x + (\\kappa u_y)_y + \\Psi(x,y).\n",
    "$$\n",
    "If $u_t = 0$ we then have\n",
    "$$\n",
    "    (\\kappa u_x)_x + (\\kappa u_y)_y = -\\Psi(x,y).\n",
    "$$"
   ]
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   "source": [
    "If we assume $\\kappa$ to be constant we can rewrite this equation as\n",
    "$$\n",
    "    u_{xx} + u_{yy} = -\\frac{\\Psi}{\\kappa} = f(x,y)\n",
    "$$\n",
    "which is the higher dimensional form of Poisson's equation (and Laplace's equation if $f(x,y) = 0$).\n",
    "\n",
    "We also must specify a domain $\\Omega$ and boundary conditions as before.  Now the boundary conditions will have $x$ and $y$ dependence so will need to be handled a bit differently than before."
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   "source": [
    "## 5-Point Stencil\n",
    "\n",
    "Extending our work in one-dimension to two we can construct the 5-point stencil for the Laplacian operator as\n",
    "$$\n",
    "    \\nabla^2 u \\approx \\frac{U_{i-1,j} - 2 U_{i,j} + U_{i+1,j}}{\\Delta x^2} + \\frac{U_{i,j-1} - 2 U_{i,j} + U_{i,j+1}}{\\Delta y^2}.\n",
    "$$"
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   "source": [
    "If $\\Delta x = \\Delta y$ we can simplify this expression to\n",
    "$$\n",
    "    \\nabla^2 u \\approx \\frac{U_{i-1,j} + U_{i+1,j} - 4 U_{i,j}  + U_{i,j-1} + U_{i,j+1}}{\\Delta x^2}\n",
    "$$"
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   "source": [
    "We can the approximate the solution to the Poisson equation as\n",
    "$$\n",
    "    \\frac{U_{i-1,j} + U_{i+1,j} - 4 U_{i,j}  + U_{i,j-1} + U_{i,j+1}}{\\Delta x^2} = f_{i,j}\n",
    "$$\n",
    "which leads to $m^2$ equations for $m^2$ unknowns."
   ]
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   "source": [
    "## Ordering of Unknowns"
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   "source": [
    "We can think of our two-dimensional problem as laying on a grid as in the following figure.  Here the unknows are in red and the boundaries are in blue:\n",
    "![2D Grid](./images/2d_grid.png)"
   ]
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    "We can think of the grid as being a matrix like\n",
    "$$\n",
    "    U = \\begin{bmatrix}\n",
    "        U_{1, m}   & U_{2, m}   & \\cdots & U_{m-1, m}   & U_{m,m}    \\\\\n",
    "        U_{1, m-1} & U_{2, m-1} & \\cdots & U_{m-1, m-1} & U_{m, m-1} \\\\\n",
    "        \\vdots & & & & \\vdots \\\\\n",
    "        U_{1, 2} & U_{2, 2}     & \\cdots & U_{m-1, 2}   & U_{m, 2}   \\\\\n",
    "        U_{1, 1} & U_{2, 1}     & \\cdots & U_{m-1, 1}   & U_{m, 1}   \\\\\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "It is important to note that this labeling does not match our usual matrix notation!"
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    "When we had a single dimension the ordering of the unknowns (and equations) was pretty clear but now we have a number of choices to make.  Perhaps the obvious choice is to break up the 2d array into rows and append each to the end of the previous one.  This leads to the ordering\n",
    "$$\n",
    "    U = [U_{1, 1}, U_{2, 1}, \\ldots, U_{m-1,1}, U_{m, 1}, U_{1, 2}, U_{22},\\ldots, U_{m-1,m}, U_{mm}]\n",
    "$$\n",
    "\n",
    "See if you can figure out the structure of the matrix $A$ now with this ordering."
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    "This leads to a matrix of the form\n",
    "$$\n",
    "    A = \\frac{1}{\\Delta x^2} \\begin{bmatrix}\n",
    "        T & I \\\\\n",
    "        I & T & I \\\\\n",
    "          & I & T & I \\\\\n",
    "          &   & \\ddots & \\ddots & \\ddots \\\\\n",
    "          &   &        & I & T & I \\\\\n",
    "          &   &        &   & I & T\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "where the matrix $T$ is\n",
    "$$\n",
    "    T = \\begin{bmatrix}\n",
    "        -4 & 1 \\\\\n",
    "        1 & -4 & 1 \\\\\n",
    "          & 1 & -4 & 1 \\\\\n",
    "          &   & \\ddots & \\ddots & \\ddots \\\\\n",
    "          &   &        & 1 & -4 & 1 \\\\\n",
    "          &   &        &   & 1 & -4\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "and $I$ is the $m \\times m$ identity matrix."
   ]
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    "Another possibility is *red-black* ordering (like a checker board).  Here we \"color\" each unknown in our grid alternating colors, each colored unknown then is grouped together.  \n",
    "\n",
    "![2D Grid Red-Black Ordering](./images/2d_grid_red_black.png)\n",
    "\n",
    "Can you figure out what the matrix looks like in this case?"
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    "For red-black ordering we are lead to the matrix\n",
    "$$\n",
    "    \\begin{bmatrix}\n",
    "        D & H \\\\\n",
    "        H^T & D\n",
    "    \\end{bmatrix} \\begin{bmatrix}\n",
    "        U_{\\text{red}} \\\\\n",
    "        U_{\\text{black}}\n",
    "    \\end{bmatrix} =\n",
    "    \\begin{bmatrix}\n",
    "        f_{\\text{red}} \\\\\n",
    "        f_{\\text{black}}\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "Here $D = -4 / \\Delta x^2 I$, a diagonal matrix of dimension $m^2 / 2$ and $H$ a *banded matrix* with 4 non-zero diagonals.\n",
    "\n",
    "Usually the goal with ordering is either to make the gaps between non-zero elements as small as possible and reduce the amount of \"fill-in\" when solving using say Gaussian elimination.  This is often taken care of automatically when using sophisticated solving routines."
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    "Solve the following PDE using the ordering of your choice:\n",
    "$$\n",
    "    \\nabla^2 u = -2\\sin x \\sin y ~~~ \\Omega = [0, 2 \\pi] \\times [0 , 2 \\pi] \\\\\n",
    "    u(x,0) = 0, u(x, 2 \\pi) = 0, u(0, y) = 0, u(2 \\pi, y) = 0.\n",
    "$$"
   ]
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      "0.00129000749498\n"
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XbpjVZBzHcZzFojbCgc6q7Z2QedhCPh7A50TkQhG5SUQ+LyJPL+3lOI7jOEMgyH/ZVfnn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\nEPTjxOkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11340a7d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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YfoK1LeRiAJeKyPUArkMaLTMP4FIAEJG3AjhVVddylbwXwIUi8nYAH0C6mHgm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\n8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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1165962d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.sparse as sparse\n",
    "import scipy.sparse.linalg as linalg\n",
    "\n",
    "# Problem specification\n",
    "m = 100\n",
    "x = numpy.linspace(0, 2.0 * numpy.pi, m + 2)\n",
    "y = numpy.linspace(0, 2.0 * numpy.pi, m + 2)\n",
    "delta_x = 2.0 * numpy.pi / (m + 1)\n",
    "\n",
    "# Construct A\n",
    "e = numpy.ones(m)\n",
    "T = sparse.spdiags([e, -4.0 * e, e], [-1, 0, 1], m, m)\n",
    "S = sparse.spdiags([e, e], [-1, 1], m, m)\n",
    "I = sparse.eye(m)\n",
    "A = sparse.kron(I, T) + sparse.kron(S, I)\n",
    "A /= delta_x**2\n",
    "\n",
    "# Right-hand-side\n",
    "X, Y = numpy.meshgrid(x[1:-1], y[1:-1])\n",
    "f = -2.0 * numpy.sin(X) * numpy.sin(Y)\n",
    "\n",
    "# Solve\n",
    "U = numpy.zeros((m+2, m+2))\n",
    "U[1:-1, 1:-1] = linalg.spsolve(A, f.reshape(m**2, order='F')).reshape((m, m), order='F')\n",
    "\n",
    "# Error\n",
    "X, Y = numpy.meshgrid(x, y)\n",
    "print numpy.linalg.norm((x[1] - x[0]) * (U - numpy.sin(X) * numpy.sin(Y)), ord=1)\n",
    "\n",
    "# Plot solution\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "sol_plot = axes.pcolor(X, Y, U, cmap=plt.get_cmap('RdBu_r'))\n",
    "axes.set_title(\"Solution u(x,y)\")\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"y\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((0.0, 2.0 * numpy.pi))\n",
    "cbar = fig.colorbar(sol_plot, ax=axes)\n",
    "cbar.set_label(\"u(x, y)\")\n",
    "\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "sol_plot = axes.pcolor(X, Y, numpy.abs(U - numpy.sin(X) * numpy.sin(Y)), cmap=plt.get_cmap('RdBu_r'))\n",
    "axes.set_title(\"Error |U - u|\")\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"y\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((0.0, 2.0 * numpy.pi))\n",
    "cbar = fig.colorbar(sol_plot, ax=axes)\n",
    "cbar.set_label(\"u(x, y)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Accuracy and Stability"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can of course (and should) ask the same questions as with the one-dimensional case, namely whether our scheme will converge.  To do this we need to consider the LTE and the stability of our method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We know that the LTE is defined as\n",
    "$$\\begin{aligned}\n",
    "    \\tau_{ij} &= \\frac{1}{\\Delta x^2} (u(x_{i-1}, y_{j}) + u(x_{i+1}, y_{j}) + u(x_{i}, y_{j-1}) + u(x_{i}, y_{j+1}) - 4 u(x_{i}, y_{j})) - f(x_i, y_j)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To compute this expression we need the Taylor series in each direction.\n",
    "\n",
    "For the x-direction:\n",
    "$$\\begin{aligned}\n",
    "    u(x_{i+1}, y_{j}) &= u(x_{i}, y_{j}) + \\Delta x u(x_{i}, y_{j})_x + \\frac{\\Delta x^2}{2} u(x_{i}, y_{j})_{xx} + \\frac{\\Delta x^3}{6} u(x_{i}, y_{j})_{xxx} + \\frac{\\Delta x^4}{24} u(x_{i}, y_{j})_{xxxx} + \\mathcal{O}(\\Delta x^5) \\\\\n",
    "    u(x_{i-1}, y_{j}) &= u(x_{i}, y_{j}) - \\Delta x u(x_{i}, y_{j})_x + \\frac{\\Delta x^2}{2} u(x_{i}, y_{j})_{xx} - \\frac{\\Delta^3}{6} u(x_{i}, y_{j})_{xxx} + \\frac{\\Delta x^4}{24} u(x_{i}, y_{j})_{xxxx} + \\mathcal{O}(\\Delta x^5)\n",
    "\\end{aligned}$$\n",
    "\n",
    "For the y-direction:\n",
    "$$\\begin{aligned}\n",
    "    u(x_{i}, y_{j+1}) &= u(x_{i}, y_{j}) + \\Delta y u(x_{i}, y_{j})_y + \\frac{\\Delta y^2}{2} u(x_{i}, y_{j})_{yy} + \\frac{\\Delta y^3}{6} u(x_{i}, y_{j})_{yyy} + \\frac{\\Delta y^4}{24} u(x_{i}, y_{j})_{yyyy} + \\mathcal{O}(\\Delta y^5) \\\\\n",
    "    u(x_{i}, y_{j-1}) &= u(x_{i}, y_{j}) - \\Delta y u(x_{i}, y_{j})_y + \\frac{\\Delta y^2}{2} u(x_{i}, y_{j})_{yy} - \\frac{\\Delta y^3}{6} u(x_{i}, y_{j})_{yyy} + \\frac{\\Delta y^4}{24} u(x_{i}, y_{j})_{yyyy} + \\mathcal{O}(\\Delta y^5)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Also using the Taylor expansions in the y-direction we can write the LTE as\n",
    "$$\n",
    "    \\tau_{ij} =  \\frac{1}{12} \\Delta x^2 (u_{xxxx} + u_{yyyy}) + \\mathcal{O}(\\Delta x^4).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The linear system for the LTE then has the form\n",
    "$$\n",
    "    A^{\\Delta x} E^{\\Delta x} = -\\tau^{\\Delta x}\n",
    "$$\n",
    "where now $A$ is the discretization we wrote before.  Note that the ordering of the equations does not matter when considering the error."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For the stability in the 2-norm we again can consider the eigenvalues of the system above.  The eigenvalues are\n",
    "$$\n",
    "    \\lambda_{pq} = \\frac{2}{\\Delta x^2} ((\\cos(p \\pi \\Delta x) - 1) + (\\cos(q \\pi \\Delta x) - 1))\n",
    "$$\n",
    "with corresponding eigenvectors\n",
    "$$\n",
    "    v^{p,q}_{ij} = \\sin(p \\pi i \\Delta x) \\sin(q \\pi j \\Delta x).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Since the eigenvalues are strictly negative ($A$ is in fact negative definite) the closest one to the origin is\n",
    "$$\n",
    "    \\lambda_{11} = -2 \\pi^2 + \\mathcal{O}(\\Delta x^2)\n",
    "$$\n",
    "leading to the spectral radius\n",
    "$$\n",
    "    \\rho((A^{\\Delta x})^{-1}) = \\frac{1}{\\lambda_{11}} \\approx -\\frac{1}{2 \\pi^2}.\n",
    "$$\n",
    "We can use this bound on $A^{-1}$ then to show stability and hence convergence of the discretization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "A similar and useful quantity to consider is the *condition number* of the matrix $A$.  Recall that this can be defined as\n",
    "$$\n",
    "    \\kappa(A) = ||A|| ||A^{-1}||.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In the 2-norm we already know some information about $A^{-1}$ but we can use our expressions from above to also find the spectral radius of $A$.  The largest eigenvalue there is \n",
    "$$\n",
    "    \\lambda_{mm} \\approx -\\frac{8}{\\Delta x^2}\n",
    "$$\n",
    "leading to the condition number\n",
    "$$\n",
    "    \\kappa_2(A) = \\frac{4}{\\pi^2 \\Delta x^2} = \\mathcal{O}\\left(\\frac{1}{\\Delta x^2}\\right)\n",
    "$$\n",
    "This matrix therefore becomes more ill-conditioned as $\\Delta x \\rightarrow 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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tuoTokXE5yDk8Mg4vtoyDH/0xFXD0BwAAkzPVjv4AAAA1QFMBAAAKQVNRM61nWEPxyLgc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\nVG+zlfyLTWY2YGaPm9lXzOz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      "text/plain": [
       "<matplotlib.figure.Figure at 0x116596050>"
      ]
     },
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    }
   ],
   "source": [
    "# Compute the eigenvalues\n",
    "m = 10\n",
    "delta_x = 1.0 / float(m + 1)\n",
    "eig_vals = numpy.empty(m**2)\n",
    "for (i, p) in enumerate(numpy.arange(1, m + 1)):\n",
    "    for (j, q) in enumerate(numpy.arange(1, m + 1)):\n",
    "        eig_vals[i * m + j] = 2.0 / delta_x**2 * ((numpy.cos(p * numpy.pi * delta_x) - 1.0) +\n",
    "                                                  (numpy.cos(q * numpy.pi * delta_x) - 1.0))\n",
    "\n",
    "# Plot of the eigenvalues of the 5-point Laplacian\n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(eig_vals.real, eig_vals.imag, 'x')\n",
    "axes.grid(True)\n",
    "axes.set_xlim(numpy.amin(eig_vals), -numpy.amin(eig_vals) * 0.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## 9-Point Laplacian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Another approach to discretizing the Laplacian $\\nabla^2$ is called the 9-point stencil.  This stencil is defined as\n",
    "$$\n",
    "    \\nabla^2 u_{ij} = \\frac{1}{6 \\Delta x^2} \\left(4 u_{i-1,j} + 4 u_{i+1,j} + 4 u_{i,j-1} + 4 u_{i,j+1} + u_{i-1,j-1} + u_{i+1,j-1} + u_{i+1,j+1} + u_{i-1,j+1} -20 u_{i,j}  \\right).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can also find the LTE as before to find\n",
    "$$\n",
    "    \\tau_{ij} = \\frac{\\Delta x^2}{12} (u_{xxxx} + u_{xxyy} + u_{yyyy}) + \\mathcal{O}(\\Delta x^4).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "So what did we gain?  If we look at $\\tau_{ij}$ we can actually write the derivative terms as\n",
    "$$\n",
    "    u_{xxxx} + u_{xxyy} + u_{yyyy} = \\nabla^2 (\\nabla^2 u) \\equiv \\nabla^4 u,\n",
    "$$\n",
    "i.e. the Laplacian of the Laplacian (called the *biharmonic operator*).  If we return to the original PDE, we can directly find the form of this error by noting\n",
    "$$\\begin{aligned}\n",
    "    \\nabla^2 u &= f \\\\\n",
    "    \\nabla^2 (\\nabla^2 u) &= \\nabla^2 f\n",
    "\\end{aligned}$$\n",
    "We can therefore compute the error terms directly by taking the Laplacian of the right hand side of the original PDE!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "This also can lead to a higher-order method.  If we define\n",
    "$$\n",
    "    f_{ij} = f(x_i, y_j) + \\frac{\\Delta x^2}{12} \\nabla^2 f(x_i, y_j)\n",
    "$$\n",
    "we can cancel out the next term of the LTE!  We can even do this if we only have the function $f(x,y)$ on a set of discrete points as we can use the 5-point stencil"
   ]
  }
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