{
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    "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": "3.7.9"
    },
    "colab": {
      "name": "504_Futher Notes on Stability.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/john-s-butler-dit/Numerical-Analysis-Python/blob/master/Chapter%2005%20-%20IVP%20Consistent%20Convergence%20Stability/504_Futher%20Notes%20on%20Stability.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "z3nFEkNZvZZs"
      },
      "source": [
        "# Further Notes on Stability\n",
        "\n",
        "In this notebook I will discuss stability for a multistep methods."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "qAa3Cen9vZZu",
        "outputId": "a5254fc6-9d4c-48b5-f59a-4c75afcd64c5",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 336
        }
      },
      "source": [
        "from IPython.display import HTML\n",
        "HTML('<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/1BviXbmtXo4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen></iframe>')"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/1BviXbmtXo4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen></iframe>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aRu2sinnvZZ0"
      },
      "source": [
        "## Definition of Stability\n",
        "The stability of a numerical method is not as tangable as consistency and convergence but when you see an unstable solution it is obvious.\n",
        "\n",
        "To determine the stabilty of a multistep method we need three definitions:\n",
        "\n",
        "\n",
        "### Definition: Characteristic Equation\n",
        "Associated with the difference equation \n",
        "\\begin{equation} w_0=\\alpha \\ \\ \\ w_1=\\alpha_1 \\ \\ \\ ... \\ \\ \\ w_{m-1}=\\alpha_{m-1} \\end{equation}\n",
        "\\begin{equation}w_{i+1} = a_{m-1}w_{i}+a_{m-2}w_{i-1}+...+a_{0}w_{i+1-m} +hF(t_i,h,w_{i+1},...,w_{i+1-m}),\\end{equation}\n",
        "is the __characteristic equation__ given by\n",
        "\\begin{equation}\\lambda^{m} - a_{m-1}\\lambda^{m-1}-a_{m-2}\\lambda^{m-2}-...-a_{0} =0. \\end{equation}\n",
        "\n",
        "### Definition: Root Condition \n",
        "\n",
        "Let $\\lambda_1,...,\\lambda_m$ denote the roots of the that characteristic equation\n",
        "\\begin{equation}\\lambda^{m} - a_{m-1}\\lambda^{m-1}-a_{m-2}\\lambda^{m-2}-...-a_{0} =0 \\end{equation}\n",
        "associated with the multi-step difference method\n",
        "\\begin{equation} w_0=\\alpha \\ \\ \\ w_1=\\alpha_1 \\ \\ \\ ... \\ \\ \\ w_{m-1}=\\alpha_{m-1} \\end{equation}\n",
        "\\begin{equation} w_{i+1} = a_{m-1}w_{i}+a_{m-2}w_{i-1}+...+a_{0}w_{i+1-m} +hF(t_i,h,w_{i+1},...,w_{i+1-m}),\\end{equation}\n",
        "If $|\\lambda_{i}|\\leq 1$ for each $i=1,...,m$ and all roots with absolute value 1\n",
        "are simple roots then the difference equation is said to satisfy the __root condition__.\n",
        "\n",
        "### Definition: Stability\n",
        "1. Methods that satisfy the root condition and have $\\lambda=1$ as the only root \n",
        "of the characteristic equation of magnitude one and all other roots are 0 are called __strongly stable__;\n",
        "2. Methods that satisfy the root condition and have more than one distinct root\n",
        "with magnitude one are called __weakly stable__;\n",
        "3. Methods that do not satisfy the root condition are called __unstable__.\n",
        "\n",
        "All one step methods, Adams-Bashforth and Adams-Moulton methods are all stongly stable."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "m7_2tkuFvZZ1"
      },
      "source": [
        "## LIBRARIES\n",
        "\n",
        "import numpy as np\n",
        "\n",
        "%matplotlib inline\n",
        "import matplotlib.pyplot as plt # side-stepping mpl backend\n",
        "import matplotlib.gridspec as gridspec # subplots\n",
        "import warnings\n",
        "\n",
        "warnings.filterwarnings(\"ignore\")"
      ],
      "execution_count": 7,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Y0AsWyENvZZ4"
      },
      "source": [
        "## Initial Value Problem\n",
        "To illustrate stability of a method we will use the given the non-linear Initial Value Problem, \n",
        "\\begin{equation} y^{'}=-0.5y^2, \\ \\ y(0)=3, \\ \\ \\  0\\leq t \\leq 10.\\end{equation}\n",
        "For the methods we will use $N=100$, which give $h=\\frac{1}{10}$ and \n",
        "\\begin{equation}t_i=0+0.1 i,\\end{equation}\n",
        "where $i=0,...100.$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "wxEoRSeGvZZ5"
      },
      "source": [
        "tau=-0.5\n",
        "N=100\n",
        "h=1/N\n",
        "\n",
        "time=np.linspace(0,10,N)\n",
        "## INITIAL CONDITIONS\n",
        "NS=np.ones(N)\n",
        "NS1=np.ones(N)\n",
        "NS2=np.ones(N)"
      ],
      "execution_count": 8,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "KLUPHRwmvZZ8"
      },
      "source": [
        "We will apply the three following methods to the above initial value problem:\n",
        "1. A stable method,\n",
        " \\begin{equation} w_{i+1}=w_{i}+\\frac{h}{2}[3(-0.5w^2_{i})-(-0.5w^2_{i-1})], \\end{equation}\n",
        "    with the characteristic equation\n",
        "\\begin{equation}\\lambda^2=\\lambda,\\end{equation}\n",
        "    which satisfies the root condition $\\lambda=1$ and $\\lambda=0$, hence it is strongly stable.\n",
        "2. A weakly stable method\n",
        "\\begin{equation} w_{i+1}=1.0w_{i-2}+h[3(-0.5w^2_{i})-(-0.5w^2_{i-1})], \\end{equation}\n",
        "    with the characteristic equation\n",
        "  \\begin{equation}\\lambda^4=1,\\end{equation}\n",
        "    which does  satisfies the root condition  with roots $\\lambda=\\pm1$ and $\\lambda=\\pm \\sqrt{-1}$ but as it has more than one root $|\\lambda|=1$ it is weakly stable.\n",
        "3. An unstable method\n",
        " \\begin{equation} w_{i+1}=1.01w_{i}+\\frac{h}{2}[3(-0.5w^2_{i})-(-0.5w^2_{i-1})] \\end{equation}\n",
        "    with the characteristic equation\n",
        "   \\begin{equation}\\lambda^2=1.01\\lambda,\\end{equation}\n",
        "    which does not satisfies the root condition, as it has the roots $\\lambda=1.01$ and $\\lambda=0$, hence it is unstable."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "GgFH94LZvZZ9"
      },
      "source": [
        "# INITIAL SOLUTIONS ONE STEP METHOD\n",
        "for i in range (0,3):\n",
        "    NS[i+1]=NS[i]+h*tau*(NS[i]*NS[i])\n",
        "    NS1[i+1]=NS[i+1]#+h*tau*(-NS[i-1]*NS[i-1])\n",
        "    NS2[i+1]=NS[i+1]#+h*tau*(-NS[i-1]*NS[i-1])\n",
        "    \n",
        "\n",
        "# MULTISTEP METHODS\n",
        "for i in range (3,N-1):\n",
        "    NS[i+1]=NS[i]+h/2*tau*(3*NS[i]*NS[i]-NS[i-1]*NS[i-1])\n",
        "    NS1[i+1]=NS1[i-2]+h*tau*(3*NS1[i]*NS1[i]-NS1[i-1]*NS1[i-1])\n",
        "    NS2[i+1]=1.01*NS2[i]+h/2*tau*(3*NS2[i]*NS2[i]-NS2[i-1]*NS2[i-1])\n",
        "                                                                 "
      ],
      "execution_count": 9,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FB48aypZvZZ_"
      },
      "source": [
        "## Results\n",
        "The below plot shows the solutions for the stable method (left), the weakly stable method (middle) and unstable method (right).\n",
        "1. The stable results show a monotonically decreasing function.  \n",
        "2. The weakly stable results have a oscilation which is a by product of the method being unstable that is not part of the exact solution.\n",
        "3. The unstable method is nothing like stable solution it is monotonically increasing after the initial conditons. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "scrolled": false,
        "id": "fnwvfQWKvZaA",
        "outputId": "eca7f30f-2329-40db-9a38-3a932bfe71c8",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 302
        }
      },
      "source": [
        "fig = plt.figure(figsize=(12,4))\n",
        "# --- left hand plot\n",
        "ax = fig.add_subplot(1,3,1)\n",
        "plt.plot(time,NS,color='red')\n",
        "#ax.legend(loc='best')\n",
        "plt.title('Stable')\n",
        "ax = fig.add_subplot(1,3,2)\n",
        "plt.plot(time,NS1)\n",
        "plt.title('Weakly Stable')\n",
        "ax = fig.add_subplot(1,3,3)\n",
        "plt.plot(time,NS2)\n",
        "plt.title('Not Stable')\n",
        "fig.suptitle(r\"Stability $y'=-0.5y^2, \\ \\  y(0)=1$\", fontsize=20)\n",
        "plt.tight_layout()\n",
        "plt.subplots_adjust(top=0.85)"
      ],
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 864x288 with 3 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "collapsed": true,
        "id": "xqYLscbgvZaC"
      },
      "source": [
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "VGckxWgYvZaC"
      },
      "source": [
        ""
      ],
      "execution_count": 10,
      "outputs": []
    }
  ]
}