ketch/stegotons.ipynb

## stegotons.ipynb
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     "source": [
      "# Stegotons: solitary waves arising in non-dispersive periodic media\n",
      "A fascinating new class of solitary waves was discovered in [a 2003 paper by R.J. LeVeque and D. Yong](http://faculty.washington.edu/rjl/pubs/solitary/40815.pdf).  Solitary waves usually appear as solutions of nonlinear dispersive wave equations, like the KdV or NLS equations.  But the waves discovered by LeVeque and Yong arise in a system of nonlinear wave equations **with no dispersion!**  The nonlinear elasticity equations they investigated are:\n",
      "\\begin{align}\n",
      "\\epsilon_t(x,t) - u_x(x,t) & = 0 \\\\\n",
      "(\\rho(x) u(x,t))_t - \\sigma(\\epsilon(x,t),x)_x & = 0.\n",
      "\\end{align}\n",
      "They took the density $\\rho(x)$ and bulk modulus $K(x)$ to be periodic functions and the stress strain relation $\\sigma(\\epsilon,x)$ nonlinear (they used an exponential function, but any nonlinear function will work).\n",
      "\n",
      "If $\\rho$ and $K$ are chosen so that the impedance $Z = \\sqrt{\\rho K}$ varies in space, then waves undergo reflection on the fine scale.  Remarkably, the effect of these reflections is an effective behavior that mimics dispersion -- as [predicted already by Santosa & Symes in 1993](http://epubs.siam.org/doi/abs/10.1137/0151049).\n",
      "\n",
      "Here we reproduce some of their original experiments in PyClaw.  If you're interested in the simulations, all of the code is provided here and you can run it yourself.  If you're only interested in the results, feel free to just skip over the code."
     ]
    },
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     "cell_type": "code",
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     "input": [
      "from clawpack import riemann\n",
      "from clawpack import pyclaw\n",
      "import numpy as np\n",
      "\n",
      "riemann_solver = riemann.nonlinear_elasticity_fwave_1D\n",
      "solver = pyclaw.ClawSolver1D(riemann_solver)\n",
      "solver.fwave = True\n",
      "\n",
      "# Boundary conditions\n",
      "solver.bc_lower[0] = pyclaw.BC.extrap\n",
      "solver.bc_upper[0] = pyclaw.BC.extrap\n",
      "solver.aux_bc_lower[0] = pyclaw.BC.extrap\n",
      "solver.aux_bc_upper[0] = pyclaw.BC.extrap\n",
      "\n",
      "xlower=0.0; xupper=1000.0\n",
      "cells_per_layer=12; mx=int(round(xupper-xlower))*cells_per_layer\n",
      "x = pyclaw.Dimension('x',xlower,xupper,mx)\n",
      "domain = pyclaw.Domain(x)\n",
      "state = pyclaw.State(domain,solver.num_eqn,3)\n",
      "xc=state.grid.x.centers\n",
      "\n",
      "#Initialize q and aux\n",
      "KA    = 1.0; rhoA  = 1.0\n",
      "KB    = 4.0; rhoB  = 4.0\n",
      "xfrac = xc-np.floor(xc)\n",
      "\n",
      "state.aux[0,:] = rhoA*(xfrac<0.5)+rhoB*(xfrac>=0.5) #Density\n",
      "state.aux[1,:] = KA  *(xfrac<0.5)+KB  *(xfrac>=0.5) #Bulk modulus\n",
      "state.aux[2,:] = 0. # not used\n",
      "\n",
      "sigma = 0.5*np.exp(-((xc-500.)/5.)**2.)\n",
      "state.q[0,:] = np.log(sigma+1.)/state.aux[1,:]  # Strain\n",
      "state.q[1,:] = 0.                               # Momentum\n",
      "\n",
      "claw = pyclaw.Controller()\n",
      "claw.solution = pyclaw.Solution(state,domain)\n",
      "\n",
      "claw.output_style = 1\n",
      "claw.num_output_times = 100\n",
      "claw.tfinal =  550.\n",
      "claw.solver = solver\n",
      "claw.keep_copy = True\n",
      "claw.output_format = None"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Before running the simulation, let's take a quick look at the setup.  The initial condition (stored in `state.q`) is a Gaussian stress perturbation with zero velocity, while the impedance $Z(x)$ is piecewise constant and periodic.  Notice in the plot below that, though the stress is a Gaussian, the strain is discontinuous since the bulk modulus $K(x)$ is discontinuous.\n",
      "\n",
      "In each plot, the inset is a closeup of the central region."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import plotly\n",
      "py = plotly.plotly(username_or_email='DavidKetcheson', key='mgs2lgb203')\n",
      "\n",
      "data = [{'x':xc, 'y':state.q[0,:]},{'x':xc[5800:6200],'y':state.q[0,5800:6200],\"xaxis\":\"x2\",\"yaxis\":\"y2\"}]\n",
      "layout = {\"title\": \"$\\\\text{Strain }(\\epsilon)$\",'showlegend':False,    \"xaxis2\": {\"domain\": [0.6, 0.95],\"anchor\": \"y2\"},\n",
      "    \"yaxis2\":{\"domain\": [0.2, 0.6],\"anchor\": \"x2\"}}\n",
      "py.iplot(data,layout=layout)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<iframe height=\"500\" id=\"igraph\" scrolling=\"no\" seamless=\"seamless\" src=\"https://plot.ly/~DavidKetcheson/64/600/450\" width=\"650\"></iframe>"
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Z = np.sqrt(state.aux[0,:]*state.aux[1,:])\n",
      "data = [{'x':xc, 'y':Z},{'x':xc[5800:6200],'y':Z[5800:6200],\"xaxis\":\"x2\",\"yaxis\":\"y2\"}]\n",
      "layout = {\"title\": \"$\\\\text{Impedance }(Z)$\",'showlegend':False,    \"xaxis2\": {\"domain\": [0.6, 0.95],\"anchor\": \"y2\"},\n",
      "    \"yaxis2\":{\"domain\": [0.2, 0.6],\"anchor\": \"x2\"}}\n",
      "py.iplot(data,layout = layout)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<iframe height=\"500\" id=\"igraph\" scrolling=\"no\" seamless=\"seamless\" src=\"https://plot.ly/~DavidKetcheson/65/600/450\" width=\"650\"></iframe>"
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        "<IPython.core.display.HTML at 0x10279b350>"
       ]
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     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "claw.run()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "{'cflmax': 0.9014990065747429,\n",
        " 'dtmax': 0.06465763584693647,\n",
        " 'dtmin': 0.06436880097776293,\n",
        " 'numsteps': 86}"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from matplotlib import animation\n",
      "import matplotlib.pyplot as plt\n",
      "from clawpack.visclaw.JSAnimation import IPython_display\n",
      "\n",
      "fig = plt.figure(figsize=[8,4])\n",
      "ax = plt.axes(xlim=(xc[0], xc[-1]), ylim=(0, 0.4))\n",
      "line, = ax.plot([], [], lw=1)\n",
      "\n",
      "def fplot(i):\n",
      "    frame = claw.frames[i]\n",
      "    strain = frame.q[0,:]\n",
      "    line.set_data(xc, strain)\n",
      "    ax.set_title('Strain at t='+str(frame.t))\n",
      "    return line,\n",
      "\n",
      "animation.FuncAnimation(fig, fplot, frames=len(claw.frames))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
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        "FnSONdiL8Qci0lzwpOM5sQAAAABJRU5ErkJggg==\n",
        "\"></button>\n",
        "    <button onclick=\"animee373e13e6f38a9f.faster()\">+</button>\n",
        "  <form action=\"#n\" name=\"_anim_loop_selectee373e13e6f38a9f\" class=\"anim_control\">\n",
        "    <input id=\"_frame_noee373e13e6f38a9f\" type=\"textbox\" size=\"1\" onchange=\"animee373e13e6f38a9f.set_frame(parseInt(this.value));\" onpaste=\"this.onchange();\" oninput=\"this.onchange();\"></input>\n",
        "    <input type=\"radio\" name=\"state\" value=\"once\" > Once </input>\n",
        "    <input type=\"radio\" name=\"state\" value=\"loop\" checked> Loop </input>\n",
        "    <input type=\"radio\" name=\"state\" value=\"reflect\" > Reflect </input>\n",
        "  </form>\n",
        "</div>\n",
        "\n",
        "\n",
        "<script language=\"javascript\">\n",
        "  /* Instantiate the Animation class. */\n",
        "  /* The IDs given should match those used in the template above. */\n",
        "  (function() {\n",
        "    var img_id = \"_anim_imgee373e13e6f38a9f\";\n",
        "    var slider_id = \"_anim_slideree373e13e6f38a9f\";\n",
        "    var frame_id = \"_frame_noee373e13e6f38a9f\"\n",
        "    var loop_select_id = \"_anim_loop_selectee373e13e6f38a9f\";\n",
        "    var frames = new Array(0);\n",
        "    \n",
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