{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Methods for 1D Advection\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": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import scipy as sp\n",
        "import matplotlib.pyplot as plt"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Problem Description"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We will set up the problem for\n",
        "$$ u_t + u u_x = 0$$\n",
        "with periodic BC on the interval $[0,1]$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [],
      "source": [
        "c = 1.0\n",
        "T = 1.0 / c # end time"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Set up the Grid"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "- `dx` will be the grid spacing in the $x$-direction\n",
        "- `x` will be the grid coordinates\n",
        "- `xx` will be really fine grid coordinates"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [],
      "source": [
        "nx = 82\n",
        "x = np.linspace(0, 1, nx, endpoint=False)\n",
        "dx = x[1] - x[0]\n",
        "xx = np.linspace(0, 1, 1000, endpoint=False)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now define an initial condition:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [],
      "source": [
        "def f(x):\n",
        "    u = np.zeros(x.shape)\n",
        "    u[np.intersect1d(np.where(x>0.4), np.where(x<0.6))] = 1.0\n",
        "    return u"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f5f40cdd6a0>]"
            ]
          },
          "execution_count": 5,
          "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": [
        "plt.plot(x, f(x), lw=3, clip_on=False)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Setting the Time Step\n",
        "\n",
        "Have spatial grid. Now we need a time step. So define a ratio parameter $\\lambda$. Let\n",
        "$$ \\Delta t = \\Delta x \\frac{\\lambda}{c}$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "T = 1\n",
            "tsteps = 88\n",
            "    dx = 0.0121951\n",
            "    dt = 0.0113415\n",
            "lambda = 0.93\n"
          ]
        }
      ],
      "source": [
        "lmbda = 0.93 * np.sign(c)\n",
        "dt = dx * lmbda / abs(c)\n",
        "nt = int(T/dt)\n",
        "print('T = %g' % T)\n",
        "print('tsteps = %d' % nt)\n",
        "print('    dx = %g' % dx)\n",
        "print('    dt = %g' % dt)\n",
        "print('lambda = %g' % lmbda)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now make an index list, called $J$, so that we can access $J+1$ and $J-1$ easily"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "outputs": [],
      "source": [
        "J = np.arange(0, nx - 1)  # all vertices\n",
        "Jm1 = np.roll(J, 1)\n",
        "Jp1 = np.roll(J, -1)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Run and Animate\n",
        "\n",
        "Experiments:\n",
        "\n",
        "- Try different values of $\\lambda$.\n",
        "- Try all the methods."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "collapsed": false,
        "jupyter": {
          "outputs_hidden": false
        }
      },
      "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_img197d72825e60400a84a648fcdc790891\">\n",
              "  <div class=\"anim-controls\">\n",
              "    <input id=\"_anim_slider197d72825e60400a84a648fcdc790891\" type=\"range\" class=\"anim-slider\"\n",
              "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
              "           oninput=\"anim197d72825e60400a84a648fcdc790891.set_frame(parseInt(this.value));\">\n",
              "    <div class=\"anim-buttons\">\n",
              "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anim197d72825e60400a84a648fcdc790891.slower()\">\n",
              "          <i class=\"fa fa-minus\"></i></button>\n",
              "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim197d72825e60400a84a648fcdc790891.first_frame()\">\n",
              "        <i class=\"fa fa-fast-backward\"></i></button>\n",
              "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim197d72825e60400a84a648fcdc790891.previous_frame()\">\n",
              "          <i class=\"fa fa-step-backward\"></i></button>\n",
              "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim197d72825e60400a84a648fcdc790891.reverse_animation()\">\n",
              "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
              "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim197d72825e60400a84a648fcdc790891.pause_animation()\">\n",
              "          <i class=\"fa fa-pause\"></i></button>\n",
              "      <button title=\"Play\" aria-label=\"Play\" onclick=\"anim197d72825e60400a84a648fcdc790891.play_animation()\">\n",
              "          <i class=\"fa fa-play\"></i></button>\n",
              "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anim197d72825e60400a84a648fcdc790891.next_frame()\">\n",
              "          <i class=\"fa fa-step-forward\"></i></button>\n",
              "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim197d72825e60400a84a648fcdc790891.last_frame()\">\n",
              "          <i class=\"fa fa-fast-forward\"></i></button>\n",
              "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim197d72825e60400a84a648fcdc790891.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_select197d72825e60400a84a648fcdc790891\"\n",
              "          class=\"anim-state\">\n",
              "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_197d72825e60400a84a648fcdc790891\"\n",
              "             >\n",
              "      <label for=\"_anim_radio1_197d72825e60400a84a648fcdc790891\">Once</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_197d72825e60400a84a648fcdc790891\"\n",
              "             checked>\n",
              "      <label for=\"_anim_radio2_197d72825e60400a84a648fcdc790891\">Loop</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_197d72825e60400a84a648fcdc790891\"\n",
              "             >\n",
              "      <label for=\"_anim_radio3_197d72825e60400a84a648fcdc790891\">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_img197d72825e60400a84a648fcdc790891\";\n",
              "    var slider_id = \"_anim_slider197d72825e60400a84a648fcdc790891\";\n",
              "    var loop_select_id = \"_anim_loop_select197d72825e60400a84a648fcdc790891\";\n",
              "    var frames = new Array(88);\n",
              "    \n",
              "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA+gAAAPoCAYAAABNo9TkAAAAOnRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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              "TkSuQmCC\\\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",
              "        anim197d72825e60400a84a648fcdc790891 = 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": 8,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "text/plain": [
              "<Figure size 1000x1000 with 0 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "import time\n",
        "\n",
        "plotit = True\n",
        "u = f(x)\n",
        "\n",
        "fig = plt.figure(figsize=(10,10))\n",
        "plt.title('u vs x')\n",
        "line2, = plt.plot(x, u, lw=3, clip_on=False)\n",
        "        \n",
        "def timestepper(n):\n",
        "    # ETBS\n",
        "    u[J] = u[J] - lmbda * u[J] * (u[J] - u[Jm1])\n",
        "    \n",
        "    uex = f((xx - c * (n+1) * dt) % 1.0)\n",
        "        \n",
        "    line2.set_data(x, u)\n",
        "    \n",
        "    return line2\n",
        "\n",
        "from matplotlib import animation\n",
        "from IPython.display import HTML\n",
        "\n",
        "ani = animation.FuncAnimation(fig, timestepper, frames=nt, 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
}