{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Finite Volume Burgers\n",
        "\n",
        "Copyright (C) 2010-2020 Luke Olson<br>\n",
        "Copyright (C) 2020 Andreas Kloeckner\n",
        "\n",
        "<details>\n",
        "<summary>MIT License</summary>\n",
        "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
        "of this software and associated documentation files (the \"Software\"), to deal\n",
        "in the Software without restriction, including without limitation the rights\n",
        "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
        "copies of the Software, and to permit persons to whom the Software is\n",
        "furnished to do so, subject to the following conditions:\n",
        "\n",
        "The above copyright notice and this permission notice shall be included in\n",
        "all copies or substantial portions of the Software.\n",
        "\n",
        "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n",
        "IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n",
        "FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n",
        "AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n",
        "LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n",
        "OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n",
        "THE SOFTWARE.\n",
        "</details>"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import scipy as sp\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "from matplotlib import animation\n",
        "from IPython.display import HTML"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [],
      "source": [
        "def gaussian(x):\n",
        "    u = np.exp(-100 * (x - 0.25)**2)\n",
        "    return u\n",
        "\n",
        "\n",
        "def step(x):\n",
        "    u = np.zeros(x.shape)\n",
        "    for j in range(len(x)):\n",
        "        if (x[j] >= 0.6) and (x[j] <= 0.8):\n",
        "            u[j] = 1.0\n",
        "\n",
        "    return u\n",
        "\n",
        "def g1(x):\n",
        "    return 1+gaussian(x)\n",
        "\n",
        "def g2(x):\n",
        "    return 1+gaussian(x) + step(x)\n",
        "\n",
        "g = g1"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "tsteps = 250\n",
            "    dx = 0.00609756\n",
            "lambda = 0.95\n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f3b43472120>]"
            ]
          },
          "execution_count": 3,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "nx = 164\n",
        "x = np.linspace(0, 1, nx, endpoint=False)\n",
        "dx = x[1] - x[0]\n",
        "xx = np.linspace(0, 1, 1000, endpoint=False)\n",
        "\n",
        "lmbda = 0.95\n",
        "nt = 250\n",
        "print('tsteps = %d' % nt)\n",
        "print('    dx = %g' % dx)\n",
        "print('lambda = %g' % lmbda)\n",
        "\n",
        "J = np.arange(0, nx)  # all vertices\n",
        "Jm1 = np.roll(J, 1)\n",
        "Jp1 = np.roll(J, -1)\n",
        "\n",
        "plt.plot(x, g(x))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Plot the solution:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {},
      "outputs": [],
      "source": [
        "if 1:\n",
        "    # Burgers\n",
        "    def f(u):\n",
        "        return u**2/2\n",
        "    def fprime(u):\n",
        "        return u\n",
        "else:\n",
        "    # advection\n",
        "    def f(u):\n",
        "        return u\n",
        "    def fprime(u):\n",
        "        return 1+0*u\n",
        "\n",
        "steps_per_frame = 2"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Part I: Lax-Friedrichs\n",
        "\n",
        "Implement `rhs` for a Lax-Friedrichs flux:\n",
        "\n",
        "Recall (local) Lax-Friedrichs:\n",
        "$$\n",
        "f^{\\ast} (u^{_-}, u^+) = \\frac{f (u^-) + f (u^+)}{2} - \\frac{\\alpha}{2} (u^+ - u^-) \n",
        "\\quad\\text{with}\\quad\n",
        "\\alpha = \\max \\left( |f' (u^-)|, |f' (u^+)| \\right).$$\n",
        "Recall FV:\n",
        "$$ \\bar{u}_{j,\\ell+1} = \\bar{u}_{j,\\ell} - \\frac{h_t}{h_x} (f (u_{j + 1 / 2,\\ell}) -\n",
        "   f (u_{j - 1 / 2,\\ell})) . $$\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [],
      "source": [
        "def rhs(u):\n",
        "    uplus = u[Jp1]\n",
        "    uminus = u[J]\n",
        "    alpha = np.maximum(np.abs(fprime(uplus)), np.abs(fprime(uminus)))\n",
        "    # right-looking, between J and Jp1\n",
        "    fluxes = (\n",
        "        (f(uplus)+f(uminus))/2\n",
        "        - alpha/2*(uplus-uminus)\n",
        "        )\n",
        "    return - 1/dx*(fluxes[J]-fluxes[Jm1])"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/html": [
              "\n",
              "<link rel=\"stylesheet\"\n",
              "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css\">\n",
              "<script language=\"javascript\">\n",
              "  function isInternetExplorer() {\n",
              "    ua = navigator.userAgent;\n",
              "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
              "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
              "  }\n",
              "\n",
              "  /* Define the Animation class */\n",
              "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
              "    this.img_id = img_id;\n",
              "    this.slider_id = slider_id;\n",
              "    this.loop_select_id = loop_select_id;\n",
              "    this.interval = interval;\n",
              "    this.current_frame = 0;\n",
              "    this.direction = 0;\n",
              "    this.timer = null;\n",
              "    this.frames = new Array(frames.length);\n",
              "\n",
              "    for (var i=0; i<frames.length; i++)\n",
              "    {\n",
              "     this.frames[i] = new Image();\n",
              "     this.frames[i].src = frames[i];\n",
              "    }\n",
              "    var slider = document.getElementById(this.slider_id);\n",
              "    slider.max = this.frames.length - 1;\n",
              "    if (isInternetExplorer()) {\n",
              "        // switch from oninput to onchange because IE <= 11 does not conform\n",
              "        // with W3C specification. It ignores oninput and onchange behaves\n",
              "        // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
              "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
              "        slider.setAttribute('oninput', null);\n",
              "    }\n",
              "    this.set_frame(this.current_frame);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.get_loop_state = function(){\n",
              "    var button_group = document[this.loop_select_id].state;\n",
              "    for (var i = 0; i < button_group.length; i++) {\n",
              "        var button = button_group[i];\n",
              "        if (button.checked) {\n",
              "            return button.value;\n",
              "        }\n",
              "    }\n",
              "    return undefined;\n",
              "  }\n",
              "\n",
              "  Animation.prototype.set_frame = function(frame){\n",
              "    this.current_frame = frame;\n",
              "    document.getElementById(this.img_id).src =\n",
              "            this.frames[this.current_frame].src;\n",
              "    document.getElementById(this.slider_id).value = this.current_frame;\n",
              "  }\n",
              "\n",
              "  Animation.prototype.next_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.previous_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.first_frame = function()\n",
              "  {\n",
              "    this.set_frame(0);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.last_frame = function()\n",
              "  {\n",
              "    this.set_frame(this.frames.length - 1);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.slower = function()\n",
              "  {\n",
              "    this.interval /= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.faster = function()\n",
              "  {\n",
              "    this.interval *= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_forward = function()\n",
              "  {\n",
              "    this.current_frame += 1;\n",
              "    if(this.current_frame < this.frames.length){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.first_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.last_frame();\n",
              "        this.reverse_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.last_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_reverse = function()\n",
              "  {\n",
              "    this.current_frame -= 1;\n",
              "    if(this.current_frame >= 0){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.last_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.first_frame();\n",
              "        this.play_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.first_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.pause_animation = function()\n",
              "  {\n",
              "    this.direction = 0;\n",
              "    if (this.timer){\n",
              "      clearInterval(this.timer);\n",
              "      this.timer = null;\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.play_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = 1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_forward();\n",
              "    }, this.interval);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.reverse_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = -1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_reverse();\n",
              "    }, this.interval);\n",
              "  }\n",
              "</script>\n",
              "\n",
              "<style>\n",
              ".animation {\n",
              "    display: inline-block;\n",
              "    text-align: center;\n",
              "}\n",
              "input[type=range].anim-slider {\n",
              "    width: 374px;\n",
              "    margin-left: auto;\n",
              "    margin-right: auto;\n",
              "}\n",
              ".anim-buttons {\n",
              "    margin: 8px 0px;\n",
              "}\n",
              ".anim-buttons button {\n",
              "    padding: 0;\n",
              "    width: 36px;\n",
              "}\n",
              ".anim-state label {\n",
              "    margin-right: 8px;\n",
              "}\n",
              ".anim-state input {\n",
              "    margin: 0;\n",
              "    vertical-align: middle;\n",
              "}\n",
              "</style>\n",
              "\n",
              "<div class=\"animation\">\n",
              "  <img id=\"_anim_img9ac3506bd11c4f729b27fd6427f3fa80\">\n",
              "  <div class=\"anim-controls\">\n",
              "    <input id=\"_anim_slider9ac3506bd11c4f729b27fd6427f3fa80\" type=\"range\" class=\"anim-slider\"\n",
              "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
              "           oninput=\"anim9ac3506bd11c4f729b27fd6427f3fa80.set_frame(parseInt(this.value));\">\n",
              "    <div class=\"anim-buttons\">\n",
              "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.slower()\">\n",
              "          <i class=\"fa fa-minus\"></i></button>\n",
              "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.first_frame()\">\n",
              "        <i class=\"fa fa-fast-backward\"></i></button>\n",
              "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.previous_frame()\">\n",
              "          <i class=\"fa fa-step-backward\"></i></button>\n",
              "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.reverse_animation()\">\n",
              "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
              "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.pause_animation()\">\n",
              "          <i class=\"fa fa-pause\"></i></button>\n",
              "      <button title=\"Play\" aria-label=\"Play\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.play_animation()\">\n",
              "          <i class=\"fa fa-play\"></i></button>\n",
              "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.next_frame()\">\n",
              "          <i class=\"fa fa-step-forward\"></i></button>\n",
              "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.last_frame()\">\n",
              "          <i class=\"fa fa-fast-forward\"></i></button>\n",
              "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim9ac3506bd11c4f729b27fd6427f3fa80.faster()\">\n",
              "          <i class=\"fa fa-plus\"></i></button>\n",
              "    </div>\n",
              "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select9ac3506bd11c4f729b27fd6427f3fa80\"\n",
              "          class=\"anim-state\">\n",
              "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_9ac3506bd11c4f729b27fd6427f3fa80\"\n",
              "             >\n",
              "      <label for=\"_anim_radio1_9ac3506bd11c4f729b27fd6427f3fa80\">Once</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_9ac3506bd11c4f729b27fd6427f3fa80\"\n",
              "             checked>\n",
              "      <label for=\"_anim_radio2_9ac3506bd11c4f729b27fd6427f3fa80\">Loop</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_9ac3506bd11c4f729b27fd6427f3fa80\"\n",
              "             >\n",
              "      <label for=\"_anim_radio3_9ac3506bd11c4f729b27fd6427f3fa80\">Reflect</label>\n",
              "    </form>\n",
              "  </div>\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_img9ac3506bd11c4f729b27fd6427f3fa80\";\n",
              "    var slider_id = \"_anim_slider9ac3506bd11c4f729b27fd6427f3fa80\";\n",
              "    var loop_select_id = \"_anim_loop_select9ac3506bd11c4f729b27fd6427f3fa80\";\n",
              "    var frames = new Array(125);\n",
              "    \n",
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              "3Hbg8akAABiAY+gAABiAoAMAYACCDgCAAQg6AAAGIOgAABiAoAMAYACCDgCAAQg6AAAGIOgAABiA\\\n",
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              "AQg6AAAGIOgAABjg/wMSrOIRv8+BpwAAAABJRU5ErkJggg==\\\n",
              "\"\n",
              "\n",
              "\n",
              "    /* set a timeout to make sure all the above elements are created before\n",
              "       the object is initialized. */\n",
              "    setTimeout(function() {\n",
              "        anim9ac3506bd11c4f729b27fd6427f3fa80 = new Animation(frames, img_id, slider_id, 29.0,\n",
              "                                 loop_select_id);\n",
              "    }, 0);\n",
              "  })()\n",
              "</script>\n"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "execution_count": 6,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "text/plain": [
              "<Figure size 500x500 with 0 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "u = g(x)\n",
        "\n",
        "def timestepper(n):\n",
        "    for i in range(steps_per_frame):\n",
        "        dt = dx*lmbda/np.max(fprime(np.abs(u)))\n",
        "        u[:] = u + dt*rhs(u)\n",
        "                       \n",
        "    line.set_data(x, u)\n",
        "    return line\n",
        "\n",
        "fig = plt.figure(figsize=(5,5))\n",
        "line, = plt.plot(x, u)\n",
        "\n",
        "ani = animation.FuncAnimation(\n",
        "    fig, timestepper,\n",
        "    frames=nt//steps_per_frame,\n",
        "    interval=30)\n",
        "html = HTML(ani.to_jshtml())\n",
        "plt.clf()\n",
        "html"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3 (ipykernel)",
      "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": "3.14.3"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 4
}