{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "###### Content under Creative Commons Attribution license CC-BY 4.0, code under MIT license © 2015 L.A. Barba, C.D. Cooper, G.F. Forsyth. Based on [CFD Python](https://github.com/barbagroup/CFDPython), © 2013 L.A. Barba, also under CC-BY license." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Relax and hold steady" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is **Module 5** of the open course [**\"Practical Numerical Methods with Python\"**](http://openedx.seas.gwu.edu/courses/GW/MAE6286/2014_fall/about), titled *\"Relax and hold steady: elliptic problems\"*. \n", "If you've come this far in the [#numericalmooc](https://twitter.com/hashtag/numericalmooc) ride, it's time to stop worrying about **time** and relax. \n", "\n", "So far, you've learned to solve problems dominated by convection—where solutions have a directional bias and can form shocks—in [Module 3](http://nbviewer.ipython.org/github/numerical-mooc/numerical-mooc/tree/master/lessons/03_wave/): *\"Riding the Wave.\"* In [Module 4](http://nbviewer.ipython.org/github/numerical-mooc/numerical-mooc/tree/master/lessons/04_spreadout/) (*\"Spreading Out\"*), we explored diffusion-dominated problems—where solutions spread in all directions. But what about situations where solutions are steady?\n", "\n", "Many problems in physics have no time dependence, yet are rich with physical meaning: the gravitational field produced by a massive object, the electrostatic potential of a charge distribution, the displacement of a stretched membrane and the steady flow of fluid through a porous medium ... all these can be modeled by **Poisson's equation**:\n", "\n", "\\begin{equation}\n", "\\nabla^2 u = f\n", "\\end{equation}\n", "\n", "where the unknown $u$ and the known $f$ are functions of space, in a domain $\\Omega$. To find the solution, we require boundary conditions. These could be Dirichlet boundary conditions, specifying the value of the solution on the boundary,\n", "\n", "\\begin{equation}\n", "u = b_1 \\text{ on } \\partial\\Omega,\n", "\\end{equation}\n", "\n", "or Neumann boundary conditions, specifying the normal derivative of the solution on the boundary,\n", "\n", "\\begin{equation}\n", "\\frac{\\partial u}{\\partial n} = b_2 \\text{ on } \\partial\\Omega.\n", "\\end{equation}\n", "\n", "A boundary-value problem consists of finding $u$, given the above information. Numerically, we can do this using *relaxation methods*, which start with an initial guess for $u$ and then iterate towards the solution. Let's find out how!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Laplace's equation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The particular case of $f=0$ (homogeneous case) results in Laplace's equation:\n", "\n", "\\begin{equation}\n", "\\nabla^2 u = 0\n", "\\end{equation}\n", "\n", "For example, the equation for steady, two-dimensional heat conduction is:\n", "\n", "\\begin{equation}\n", "\\frac{\\partial ^2 T}{\\partial x^2} + \\frac{\\partial ^2 T}{\\partial y^2} = 0\n", "\\end{equation}\n", "\n", "This is similar to the model we studied in [lesson 3](http://nbviewer.ipython.org/github/numerical-mooc/numerical-mooc/blob/master/lessons/04_spreadout/04_03_Heat_Equation_2D_Explicit.ipynb) of **Module 4**, but without the time derivative: i.e., for a temperature $T$ that has reached steady state. The Laplace equation models the equilibrium state of a system under the supplied boundary conditions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The study of solutions to Laplace's equation is called *potential theory*, and the solutions themselves are often potential fields. Let's use $p$ from now on to represent our generic dependent variable, and write Laplace's equation again (in two dimensions):\n", "\n", "\\begin{equation}\n", "\\frac{\\partial ^2 p}{\\partial x^2} + \\frac{\\partial ^2 p}{\\partial y^2} = 0\n", "\\end{equation}\n", "\n", "Like in the diffusion equation of the previous course module, we discretize the second-order derivatives with *central differences*. You should be able to write down a second-order central-difference formula by heart now! On a two-dimensional Cartesian grid, it gives:\n", "\n", "\\begin{equation}\n", "\\frac{p_{i+1, j} - 2p_{i,j} + p_{i-1,j} }{\\Delta x^2} + \\frac{p_{i,j+1} - 2p_{i,j} + p_{i, j-1} }{\\Delta y^2} = 0\n", "\\end{equation}\n", "\n", "When $\\Delta x = \\Delta y$, we end up with the following equation:\n", "\n", "\n", "\\begin{equation}\n", "p_{i+1, j} + p_{i-1,j} + p_{i,j+1} + p_{i, j-1}- 4 p_{i,j} = 0\n", "\\end{equation}\n", "\n", "This tells us that the Laplacian differential operator at grid point $(i,j)$ can be evaluated discretely using the value of $p$ at that point (with a factor $-4$) and the four neighboring points to the left and right, above and below grid point $(i,j)$.\n", "\n", "The stencil of the discrete Laplacian operator is shown in Figure 1. It is typically called the *five-point stencil*, for obvious reasons.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Figure 1: Laplace five-point stencil." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The discrete equation above is valid for every interior point in the domain. If we write the equations for *all* interior points, we have a linear system of algebraic equations. We *could* solve the linear system directly (e.g., with Gaussian elimination), but we can be more clever than that!\n", "\n", "Notice that the coefficient matrix of such a linear system has mostly zeroes. For a uniform spatial grid, the matrix is *block diagonal*: it has diagonal blocks that are tridiagonal with $-4$ on the main diagonal and $1$ on two off-center diagonals, and two more diagonals with $1$. All of the other elements are zero. Iterative methods are particularly suited for a system with this structure, and save us from storing all those zeroes.\n", "\n", "We will start with an initial guess for the solution, $p_{i,j}^{0}$, and use the discrete Laplacian to get an update, $p_{i,j}^{1}$, then continue on computing $p_{i,j}^{k}$ until we're happy. Note that $k$ is _not_ a time index here, but an index corresponding to the number of iterations we perform in the *relaxation scheme*. \n", "\n", "At each iteration, we compute updated values $p_{i,j}^{k+1}$ in a (hopefully) clever way so that they converge to a set of values satisfying Laplace's equation. The system will reach equilibrium only as the number of iterations tends to $\\infty$, but we can approximate the equilibrium state by iterating until the change between one iteration and the next is *very* small. \n", "\n", "The most intuitive method of iterative solution is known as the [**Jacobi method**](https://en.wikipedia.org/wiki/Jacobi_method), in which the values at the grid points are replaced by the corresponding weighted averages:\n", "\n", "\n", "\\begin{equation}\n", "p^{k+1}_{i,j} = \\frac{1}{4} \\left(p^{k}_{i,j-1} + p^k_{i,j+1} + p^{k}_{i-1,j} + p^k_{i+1,j} \\right)\n", "\\end{equation}\n", "\n", "This method does indeed converge to the solution of Laplace's equation. Thank you Professor Jacobi!\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Challenge task\n", "\n", "Grab a piece of paper and write out the coefficient matrix for a discretization with 7 grid points in the $x$ direction (5 interior points) and 5 points in the $y$ direction (3 interior). The system should have 15 unknowns, and the coefficient matrix three diagonal blocks. Assume prescribed Dirichlet boundary conditions on all sides (not necessarily zero)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Boundary conditions and relaxation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Suppose we want to model steady-state heat transfer on (say) a computer chip with one side insulated (zero Neumann BC), two sides held at a fixed temperature (Dirichlet condition) and one side touching a component that has a sinusoidal distribution of temperature.\n", "We would need to solve Laplace's equation with boundary conditions like\n", "\n", "\\begin{equation}\n", " \\begin{gathered}\n", "p=0 \\text{ at } x=0\\\\\n", "\\frac{\\partial p}{\\partial x} = 0 \\text{ at } x = L\\\\\n", "p = 0 \\text{ at }y = 0 \\\\\n", "p = \\sin \\left( \\frac{\\frac{3}{2}\\pi x}{L} \\right) \\text{ at } y = H.\n", " \\end{gathered}\n", "\\end{equation}\n", "\n", "\n", "We'll take $L=1$ and $H=1$ for the sizes of the domain in the $x$ and $y$ directions.\n", "\n", "One of the defining features of elliptic PDEs is that they are \"driven\" by the boundary conditions. In the iterative solution of Laplace's equation, boundary conditions are set and **the solution relaxes** from an initial guess to join the boundaries together smoothly, given those conditions. Our initial guess will be $p=0$ everywhere. Now, let's relax!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, we import our usual smattering of libraries (plus a few new ones!)" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from matplotlib import pyplot\n", "import numpy\n", "%matplotlib inline\n", "from matplotlib import rcParams\n", "rcParams['font.family'] = 'serif'\n", "rcParams['font.size'] = 16" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To visualize 2D data, we can use [`pyplot.imshow()`](http://matplotlib.org/api/pyplot_api.html#matplotlib.pyplot.imshow), like we've done before, but a 3D plot can sometimes show a more intuitive view the solution. Or it's just prettier!\n", "\n", "Be sure to enjoy the many examples of 3D plots in the `mplot3d` section of the [Matplotlib Gallery](http://matplotlib.org/gallery.html#mplot3d). \n", "\n", "We'll import the `Axes3D` library from Matplotlib and also grab the `cm` package, which provides different colormaps for visualizing plots. " ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from mpl_toolkits.mplot3d import Axes3D\n", "from matplotlib import cm" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's define a function for setting up our plotting environment, to avoid repeating this set-up over and over again. It will save us some typing.\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def plot_3D(x, y, p):\n", " '''Creates 3D plot with appropriate limits and viewing angle\n", " \n", " Parameters:\n", " ----------\n", " x: array of float\n", " nodal coordinates in x\n", " y: array of float\n", " nodal coordinates in y\n", " p: 2D array of float\n", " calculated potential field\n", " \n", " '''\n", " fig = pyplot.figure(figsize=(11,7), dpi=100)\n", " ax = fig.gca(projection='3d')\n", " X,Y = numpy.meshgrid(x,y)\n", " surf = ax.plot_surface(X,Y,p[:], rstride=1, cstride=1, cmap=cm.viridis,\n", " linewidth=0, antialiased=False)\n", "\n", " ax.set_xlim(0,1)\n", " ax.set_ylim(0,1)\n", " ax.set_xlabel('$x$')\n", " ax.set_ylabel('$y$')\n", " ax.set_zlabel('$z$')\n", " ax.view_init(30,45)\n" ] }, { "cell_type": "markdown", "metadata": { "code_folding": [] }, "source": [ "##### Note \n", "This plotting function uses *Viridis*, a new (and _awesome_) colormap available in Matplotlib versions 1.5 and greater. If you see an error when you try to plot using cm.viridis, just update Matplotlib using conda or pip." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Analytical solution" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Laplace equation with the boundary conditions listed above has an analytical solution, given by\n", "\n", "\\begin{equation}\n", "p(x,y) = \\frac{\\sinh \\left( \\frac{\\frac{3}{2} \\pi y}{L}\\right)}{\\sinh \\left( \\frac{\\frac{3}{2} \\pi H}{L}\\right)} \\sin \\left( \\frac{\\frac{3}{2} \\pi x}{L} \\right)\n", "\\end{equation}\n", "\n", "where $L$ and $H$ are the length of the domain in the $x$ and $y$ directions, respectively." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We previously used `numpy.meshgrid` to plot our 2D solutions to the heat equation in Module 4. Here, we'll use it again as a plotting aid. Always useful, `linspace` creates 1-row arrays of equally spaced numbers: it helps for defining $x$ and $y$ axes in line plots, but now we want the analytical solution plotted for every point in our domain. To do this, we'll use in the analytical solution the 2D arrays generated by `numpy.meshgrid`." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def p_analytical(x, y):\n", " X, Y = numpy.meshgrid(x,y)\n", " \n", " p_an = numpy.sinh(1.5*numpy.pi*Y / x[-1]) /\\\n", " (numpy.sinh(1.5*numpy.pi*y[-1]/x[-1]))*numpy.sin(1.5*numpy.pi*X/x[-1])\n", " \n", " return p_an" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ok, let's try out the analytical solution and use it to test the `plot_3D` function we wrote above. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nx = 41\n", "ny = 41\n", "\n", "x = numpy.linspace(0,1,nx)\n", "y = numpy.linspace(0,1,ny)\n", "\n", "p_an = p_analytical(x,y)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Xj8ncOOPmxpG5w34z/PfBJ07BSYt3V+aaq1K0MSskdAp0Oh3hbBG6fxEBuwUN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\nO2bOnIlRo0bhn//8J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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_3D(x,y,p_an)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It worked! This is what the solution *should* look like when we're 'done' relaxing. (And isn't viridis a cool colormap?) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### How long do we iterate?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We noted above that there is no time dependence in the Laplace equation. So it doesn't make a lot of sense to use a `for` loop with `nt` iterations, like we've done before.\n", "\n", "Instead, we can use a `while` loop that continues to iteratively apply the relaxation scheme until the difference between two successive iterations is small enough. \n", "\n", "But how small is small enough? That's a good question. We'll try to work that out as we go along. \n", "\n", "To compare two successive potential fields, a good option is to use the [L2 norm][1] of the difference. It's defined as\n", "\n", "\\begin{equation}\n", "|\\textbf{x}| = \\sqrt{\\sum_{i=0, j=0}^k \\left|p^{k+1}_{i,j} - p^k_{i,j}\\right|^2}\n", "\\end{equation}\n", "\n", "But there's one flaw with this formula. We are summing the difference between successive iterations at each point on the grid. So what happens when the grid grows? (For example, if we're refining the grid, for whatever reason.) There will be more grid points to compare and so more contributions to the sum. The norm will be a larger number just because of the grid size!\n", "\n", "That doesn't seem right. We'll fix it by normalizing the norm, dividing the above formula by the norm of the potential field at iteration $k$. \n", "\n", "For two successive iterations, the relative L2 norm is then calculated as\n", "\n", "\\begin{equation}\n", "|\\textbf{x}| = \\frac{\\sqrt{\\sum_{i=0, j=0}^k \\left|p^{k+1}_{i,j} - p^k_{i,j}\\right|^2}}{\\sqrt{\\sum_{i=0, j=0}^k \\left|p^k_{i,j}\\right|^2}}\n", "\\end{equation}\n", "\n", "Our Python code for this calculation is a one-line function:\n", "\n", "[1]:http://en.wikipedia.org/wiki/Norm_%28mathematics%29#Euclidean_norm" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def L2_error(p, pn):\n", " return numpy.sqrt(numpy.sum((p - pn)**2)/numpy.sum(pn**2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, let's define a function that will apply Jacobi's method for Laplace's equation. Three of the boundaries are Dirichlet boundaries and so we can simply leave them alone. Only the Neumann boundary needs to be explicitly calculated at each iteration. " ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def laplace2d(p, l2_target):\n", " '''Iteratively solves the Laplace equation using the Jacobi method\n", " \n", " Parameters:\n", " ----------\n", " p: 2D array of float\n", " Initial potential distribution\n", " l2_target: float\n", " target for the difference between consecutive solutions\n", " \n", " Returns:\n", " -------\n", " p: 2D array of float\n", " Potential distribution after relaxation\n", " '''\n", " \n", " l2norm = 1\n", " pn = numpy.empty_like(p)\n", " iterations = 0\n", " while l2norm > l2_target:\n", " pn = p.copy()\n", " p[1:-1,1:-1] = .25 * (pn[1:-1,2:] + pn[1:-1, :-2] \\\n", " + pn[2:, 1:-1] + pn[:-2, 1:-1])\n", " \n", " ##Neumann B.C. along x = L\n", " p[1:-1, -1] = p[1:-1, -2]\n", " l2norm = L2_error(p, pn)\n", " \n", " return p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Rows and columns, and index order\n", "\n", "Recall that in the [2D explicit heat equation](./lessons/04_spreadout/03_Heat_Equation_2D_Explicit.ipynb) we stored data with the $y$ coordinates corresponding to the rows of the array and $x$ coordinates on the columns (this is just a code design decision!). We did that so that a plot of the 2D-array values would have the natural ordering, corresponding to the physical domain ($y$ coordinate in the vertical). \n", "\n", "We'll follow the same convention here (even though we'll be plotting in 3D, so there's no real reason), just to be consistent. Thus, $p_{i,j}$ will be stored in array format as `p[j,i]`. Don't be confused by this.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Let's relax!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The initial values of the potential field are zero everywhere (initial guess), except at the boundary: \n", "\n", "$$p = \\sin \\left( \\frac{\\frac{3}{2}\\pi x}{L} \\right) \\text{ at } y=H$$\n", "\n", "To initialize the domain, `numpy.zeros` will handle everything except that one Dirichlet condition. Let's do it!" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [], "source": [ "##variable declarations\n", "nx = 41\n", "ny = 41\n", "\n", "\n", "##initial conditions\n", "p = numpy.zeros((ny,nx)) ##create a XxY vector of 0's\n", "\n", "\n", "##plotting aids\n", "x = numpy.linspace(0,1,nx)\n", "y = numpy.linspace(0,1,ny)\n", "\n", "##Dirichlet boundary conditions\n", "p[-1,:] = numpy.sin(1.5*numpy.pi*x/x[-1])\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's visualize the initial conditions using the `plot_3D` function, just to check we've got it right.\n" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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V6jduUrOVSgXDw8O6mbHX1KzI65HM60ynxNyw5uBz0oCdI8fgQ90vSiGGnGxL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\nYuIooCOEEEIIMXH/D/MOWtlMMPNxAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_3D(x, y, p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `p` array is equal to zero everywhere, except along the boundary $y = 1$. Hopefully you can see how the relaxed solution and this initial condition are related. \n", "\n", "Now, run the iterative solver with a target L2-norm difference between successive iterations of $10^{-8}$." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "p = laplace2d(p.copy(), 1e-8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Let's make a gorgeous plot of the final field using the newly minted `plot_3D` function." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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/Ow2/uLy/MvdclaKNaSGhk6DX6wlHi1D9QQTsdmhglcyB8kbhiPKR9nugSiq1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7IoKytDUVER9u7di8GDB2Pu3Lm48847Ze8/rVq1wrRp0wLwLbwjEp/RWqFCd51Lly6J/oUmJCSg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\nCUtdBCqfz9tcOseFS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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plot_3D(x,y,p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Awesome! That looks pretty good. But we'll need more than a simple visual check, though. The \"eyeball metric\" is very forgiving!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Convergence analysis" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Convergence, Take 1" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We want to make sure that our Jacobi function is working properly. Since we have an analytical solution, what better way than to do a grid-convergence analysis? We will run our solver for several grid sizes and look at how fast the L2 norm of the difference between consecutive iterations decreases.\n", "\n", "Let's make our lives easier by writing a function to \"reset\" the initial guess for each grid so we don't have to keep copying and pasting them." ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def laplace_IG(nx):\n", " '''Generates initial guess for Laplace 2D problem for a \n", " given number of grid points (nx) within the domain [0,1]x[0,1]\n", " \n", " Parameters:\n", " ----------\n", " nx: int\n", " number of grid points in x (and implicitly y) direction\n", " \n", " Returns:\n", " -------\n", " p: 2D array of float\n", " Pressure distribution after relaxation\n", " x: array of float\n", " linspace coordinates in x\n", " y: array of float\n", " linspace coordinates in y\n", " '''\n", "\n", " ##initial conditions\n", " p = numpy.zeros((nx,nx)) ##create a XxY vector of 0's\n", "\n", " ##plotting aids\n", " x = numpy.linspace(0,1,nx)\n", " y = x\n", "\n", " ##Dirichlet boundary conditions\n", " p[:,0] = 0\n", " p[0,:] = 0\n", " p[-1,:] = numpy.sin(1.5*numpy.pi*x/x[-1])\n", " \n", " return p, x, y" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now run Jacobi's method on the Laplace equation using four different grids, with the same exit criterion of $10^{-8}$ each time. Then, we look at the error versus the grid size in a log-log plot. What do we get?" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "code_folding": [], "collapsed": false }, "outputs": [], "source": [ "nx_values = [11, 21, 41, 81]\n", "l2_target = 1e-8\n", "\n", "error = numpy.empty_like(nx_values, dtype=numpy.float)\n", "\n", "\n", "for i, nx in enumerate(nx_values):\n", " p, x, y = laplace_IG(nx)\n", " \n", " p = laplace2d(p.copy(), l2_target)\n", " \n", " p_an = p_analytical(x, y)\n", " \n", " error[i] = L2_error(p, p_an)\n", " " ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pyplot.figure(figsize=(6,6))\n", "pyplot.grid(True)\n", "pyplot.xlabel(r'$n_x$', fontsize=18)\n", "pyplot.ylabel(r'$L_2$-norm of the error', fontsize=18)\n", "\n", "pyplot.loglog(nx_values, error, color='k', ls='--', lw=2, marker='o')\n", "pyplot.axis('equal');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hmm. That doesn't look like 2nd-order convergence, but we're using second-order finite differences. *What's going on?* The culprit is the boundary conditions. Dirichlet conditions are order-agnostic (a set value is a set value), but the scheme we used for the Neumann boundary condition is 1st-order. \n", "\n", "Remember when we said that the boundaries drive the problem? One boundary that's 1st-order completely tanked our spatial convergence. Let's fix it!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2nd-order Neumann BCs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Up to this point, we have used the first-order approximation of a derivative to satisfy Neumann B.C.'s. For a boundary located at $x=0$ this reads,\n", "\n", "\\begin{equation}\n", "\\frac{p^{k+1}_{1,j} - p^{k+1}_{0,j}}{\\Delta x} = 0\n", "\\end{equation}\n", "\n", "which, solving for $p^{k+1}_{0,j}$ gives us\n", "\n", "\\begin{equation}\n", "p^{k+1}_{0,j} = p^{k+1}_{1,j}\n", "\\end{equation}\n", "\n", "Using that Neumann condition will limit us to 1st-order convergence. Instead, we can start with a 2nd-order approximation (the central-difference approximation):\n", "\n", "\\begin{equation}\n", "\\frac{p^{k+1}_{1,j} - p^{k+1}_{-1,j}}{2 \\Delta x} = 0\n", "\\end{equation}\n", "\n", "That seems problematic, since there is no grid point $p^{k}_{-1,j}$. But no matter … let's carry on. According to the 2nd-order approximation,\n", "\n", "\\begin{equation}\n", "p^{k+1}_{-1,j} = p^{k+1}_{1,j}\n", "\\end{equation}\n", "\n", "Recall the finite-difference Jacobi equation with $i=0$:\n", "\n", "\\begin{equation}\n", "p^{k+1}_{0,j} = \\frac{1}{4} \\left(p^{k}_{0,j-1} + p^k_{0,j+1} + p^{k}_{-1,j} + p^k_{1,j} \\right)\n", "\\end{equation}\n", "\n", "Notice that the equation relies on the troublesome (nonexistent) point $p^k_{-1,j}$, but according to the equality just above, we have a value we can substitute, namely $p^k_{1,j}$. Ah! We've completed the 2nd-order Neumann condition:\n", "\n", "\\begin{equation}\n", "p^{k+1}_{0,j} = \\frac{1}{4} \\left(p^{k}_{0,j-1} + p^k_{0,j+1} + 2p^{k}_{1,j} \\right)\n", "\\end{equation}\n", "\n", "That's a bit more complicated than the first-order version, but it's relatively straightforward to code." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Note \n", "\n", "Do not confuse $p^{k+1}_{-1,j}$ with p[-1]:\n", "p[-1] is a piece of Python code used to refer to the last element of a list or array named p. $p^{k+1}_{-1,j}$ is a 'ghost' point that describes a position that lies outside the actual domain." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Convergence, Take 2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can copy the previous Jacobi function and replace only the line implementing the Neumann boundary condition. \n", "\n", "##### Careful!\n", "Remember that our problem has the Neumann boundary located at $x = L$ and not $x = 0$ as we assumed in the derivation above." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def laplace2d_neumann(p, l2_target):\n", " '''Iteratively solves the Laplace equation using the Jacobi method\n", " with second-order Neumann boundary conditions\n", " \n", " Parameters:\n", " ----------\n", " p: 2D array of float\n", " Initial potential distribution\n", " l2_target: float\n", " target for the difference between consecutive solutions\n", " \n", " Returns:\n", " -------\n", " p: 2D array of float\n", " Potential distribution after relaxation\n", " '''\n", " \n", " l2norm = 1\n", " pn = numpy.empty_like(p)\n", " iterations = 0\n", " while l2norm > l2_target:\n", " pn = p.copy()\n", " p[1:-1,1:-1] = .25 * (pn[1:-1,2:] + pn[1:-1, :-2] \\\n", " + pn[2:, 1:-1] + pn[:-2, 1:-1])\n", " \n", " ##2nd-order Neumann B.C. along x = L\n", " p[1:-1,-1] = .25 * (2*pn[1:-1,-2] + pn[2:, -1] + pn[:-2, -1])\n", " \n", " l2norm = L2_error(p, pn)\n", " \n", " return p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Again, this is the exact same code as before, but now we're running the Jacobi solver with a 2nd-order Neumann boundary condition. Let's do a grid-refinement analysis, and plot the error versus the grid spacing." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [], "source": [ "nx_values = [11, 21, 41, 81]\n", "l2_target = 1e-8\n", "\n", "error = numpy.empty_like(nx_values, dtype=numpy.float)\n", "\n", "\n", "for i, nx in enumerate(nx_values):\n", " p, x, y = laplace_IG(nx)\n", " \n", " p = laplace2d_neumann(p.copy(), l2_target)\n", " \n", " p_an = p_analytical(x, y)\n", " \n", " error[i] = L2_error(p, p_an)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pyplot.figure(figsize=(6,6))\n", "pyplot.grid(True)\n", "pyplot.xlabel(r'$n_x$', fontsize=18)\n", "pyplot.ylabel(r'$L_2$-norm of the error', fontsize=18)\n", "\n", "pyplot.loglog(nx_values, error, color='k', ls='--', lw=2, marker='o')\n", "pyplot.axis('equal');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Nice! That's much better. It might not be *exactly* 2nd-order, but it's awfully close. (What is [\"close enough\"](http://ianhawke.github.io/blog/close-enough.html) in regards to observed convergence rates is a thorny question.)\n", "\n", "Now, notice from this plot that the error on the finest grid is around $0.0002$. Given this, perhaps we didn't need to continue iterating until a target difference between two solutions of $10^{-8}$. The spatial accuracy of the finite difference approximation is much worse than that! But we didn't know it ahead of time, did we? That's the \"catch 22\" of iterative solution of systems arising from discretization of PDEs." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Final word" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Jacobi method is the simplest relaxation scheme to explain and to apply. It is also the *worst* iterative solver! In practice, it is seldom used on its own as a solver, although it is useful as a smoother with multi-grid methods. As we will see in the [third lesson](http://nbviewer.ipython.org/github/numerical-mooc/numerical-mooc/blob/master/lessons/05_relax/05_03_Iterate.This.ipynb) of this module, there are much better iterative methods! But first, let's play with [Poisson's equation](http://nbviewer.ipython.org/github/numerical-mooc/numerical-mooc/blob/master/lessons/05_relax/05_02_2D.Poisson.Equation.ipynb)." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.core.display import HTML\n", "css_file = '../../styles/numericalmoocstyle.css'\n", "HTML(open(css_file, \"r\").read())" ] } ], "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": "3.5.1" } }, "nbformat": 4, "nbformat_minor": 0 }