{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Initial Value Problems for Ordinary Differential Equations, Part 1: Basic Concepts and Euler's Method\n",
    "\n",
    "**Updated on March 22**\n",
    "with added [Example 4](#example-4) and some [numerical solutions of it](#solving-example-4)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**References:**\n",
    "\n",
    "- Sections 6.1.1 *Euler's Method* in [Sauer](../references.html#Sauer)\n",
    "- Section 5.2 *Euler's Method* in [Burden&Faires](../references.html#Burden-Faires)\n",
    "- Sections 7.1 and 7.2 of [Chenney&Kincaid](../references.html#Chenney-Kincaid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "# Shortcuts for some favorite commands:\n",
    "from numpy import linspace\n",
    "from matplotlib.pyplot import figure, plot, grid, title, xlabel, ylabel, legend"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Basic ODE Initial Value Problem\n",
    "\n",
    "We consider the problem of solving (approximately) the ordinary differential equation\n",
    "\n",
    "$$\\frac{du}{dt} = f(t, u(t)), a \\leq t \\leq b$$\n",
    "\n",
    "with the *initial condition*\n",
    "\n",
    "$$u(a) = u_0$$\n",
    "\n",
    "I will follow the common custom of referring to the independent variable as \"time\".\n",
    "\n",
    "For now, $u(t)$ is real-valued, but little will change when we later let it be vector-valued (and/or complex-valued)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"IVP_notation\"></a>\n",
    "### Notation for the solution of an initial value problem\n",
    "\n",
    "Sometimes, we need to be more careful and explicit in describing the function that solves the above initial value problem;\n",
    "then the input *parameters* $a$ and $u_0 = u(a)$ will be included of the function's formula:\n",
    "\n",
    "$$u(t) = u(t; a, u_0)$$\n",
    "\n",
    "(It is standard mathematical convention to separate *parameters* like $a$ and $u_0$ from *variables* like $t$ by putting the former after a semicolon."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Three Examples (plus one more)\n",
    "\n",
    "A lot of useful intuition comes from three simple examples:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"example-1\"></a>\n",
    "### Example 1: Integration\n",
    "\n",
    "If the derivative depends only on the independent variable $t$, so that\n",
    "\n",
    "$$\n",
    "\\frac{du}{dt} = f(t), a \\leq t \\leq b\n",
    "$$\n",
    "\n",
    "the solution is given by integration:\n",
    "\n",
    "$$\n",
    "u(t) = u_0 + \\int_a^t f(s)\\, ds.\n",
    "$$\n",
    "\n",
    "In particular, with $u_0 = 0$ the value at $b$ is\n",
    "\n",
    "$$\n",
    "u(t) = \\int_a^b f(t)\\, dt,\n",
    "$$\n",
    "\n",
    "and this gives us a back-door way to use numerical methods for solving ODEs to evaluate definite integrals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"example-2\"></a>\n",
    "### Example 2: The simplest \"real\" ODE\n",
    "\n",
    "The simplest case with $u$ present in $f$ is $f(t, u) = f(u) = u$.\n",
    "But it does not hurt to add a constant, so:\n",
    "\n",
    "$$ \\frac{du}{dt} = k u,\\; k \\text{ a constant.} $$\n",
    "\n",
    "The solution is\n",
    "\n",
    "$$ u(t) = u_0 e^{kt} $$\n",
    "\n",
    "We will see that this simple example contains the essence of ideas relevant far more generally."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"example-3\"></a>\n",
    "### Example 3: A nonlinear equation, with failure of solutions\n",
    "\n",
    "In the previous examples, $f(t, u)$ is linear in $u$ (consider $t$ as fixed);\n",
    "nonlinearities can lead to more difficult behavior.\n",
    "The equation\n",
    "\n",
    "$$ \\frac{du}{dt} = u^2, \\, u(a) = u_0 $$\n",
    "\n",
    "can be solved by separation of variables — or for now you can just verify the solution\n",
    "\n",
    "$$ u(t) = \\frac{1}{T - t}, \\, T = a + 1/u_0. $$\n",
    "\n",
    "Note that if $u_0 > 0$, the only exists for $t < T$.\n",
    "(The solution is also valid for $T > 0$, but that part has no connection to the initial data at $t=a$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example 3 warns us that the IVP might not be **well-posed** when we set the interval $[a, b]$ in advance: all we can guarantee in general is that a solution exists up to some time $b$, $b > a$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"example-4\"></a>\n",
    "### Example 4 (Added March 22): A \"stiff\" equation with disparate time scales\n",
    "\n",
    "One common problem in practical situations is differential equations where some phenomena happen on a very fast time scale, but only ever at very small amplitudes, so they have very little relevance to the overall solution.\n",
    "One example is decriptions of some chemical reactions, where some reaction products (like free radicals) are producd in tiny quantities and break down very rapidly, so they change on a very fast time scale but are scarcely relevant to the overall solution.\n",
    "\n",
    "This disparity of time-sales is called *stiffness*, from the analogy of a mechanical system in which some components are very stiff and so vibrate at very high frequencies, but typically only at very small amplitudes, or very quicky damped away, so that they can often be safely described by assuming that those stiff parts are completely rigid — do not move at all.\n",
    "\n",
    "One equation that illustrates this feature is\n",
    "\n",
    "$$\n",
    "\\frac{du}{dt} = -\\sin t -K(u - \\cos t) \n",
    "$$\n",
    "\n",
    "where $K$ is large and positive.\n",
    "Its family of solutions is\n",
    "\n",
    "$$\n",
    "u(t) = \\cos t + c e^{-K t}\n",
    "$$\n",
    "with $c = u_0 - 1$ for the initial value problem $u(0) = u_0$.\n",
    "\n",
    "These all get close to $\\cos t$ quickly and then stay nearby, but with a rapid and rapidly decaying \"transient\" $c e^{-K t}$.\n",
    "\n",
    "Many of the most basic and widely use numerical methods (including Euler's Method thet we meet soon) need to use very small time steps to handle that fast transient, even when it is very small because $u_0 \\approx 1$.\n",
    "\n",
    "On the other hand there are methods that \"supress\" these transients, allowing use of larger time steps while still getting an accurate description of the main, slower, phenomena.\n",
    "The simplest of these is the *Backward Euler Method* that we will see in a later section."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Tangent Line Method, a.k.a. Euler's Method\n",
    "\n",
    "Once we know $u(t)$ (or a good approximation) at some time $t$, we also know the value of $u'(t) = f(t, u(t))$ there; in particular, we know that $u(a) = u_0$ and so $u'(a) = f(a, u_0)$.\n",
    "\n",
    "This allows us to approximate $u$ for slightly larger values of the argument (which I will call \"time\") using its tangent line:\n",
    "\n",
    "$$ u(a+h) \\approx u(a) + u'(a) h = u_0 + f(a, u_0) h  \\text{ for \"small\" $h$} $$\n",
    "\n",
    "and more generally\n",
    "\n",
    "$$ u(t+h) \\approx u(t) + f(t, u(t)) h \\text{ for \"small\" $h$} $$\n",
    "\n",
    "This leads to the simplest approximation: choose a step size $h$ determining equally spaced times $t_i = a + i h$ and define — recursively — a sequence of approximations $U_i \\approx u(t_i)$ with\n",
    "\n",
    "$$\\begin{split}\n",
    "U_0 &= u_0\n",
    "\\\\\n",
    "U_{i+1} &= U_i + h f(t_i, U_i))\n",
    "\\end{split}$$\n",
    "\n",
    "If we choose a number of time steps $n$ and set $h = (b-a)/n$ for $0 \\leq i \\leq n$,\n",
    "the second equation is needed for $0 \\leq i < n$, ending with $U_n \\approx u(t_n) = u(b)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This \"two-liner\" do not need a pseudo-code description; instead, we can go directly to a rudimentary Python function for Euler's Method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def euler(f, a, b, u_0, n=100):\n",
    "    \"\"\"Solve du/dt = f(t, u) for t in [a, b], with initial value u(a) = u_0\"\"\"\n",
    "    h = (b-a)/n\n",
    "    t = linspace(a, b, n+1)  # Note: \"n\" counts steps, so there are n+1 values for t.\n",
    "    u = np.empty_like(t)\n",
    "    u[0] = u_0\n",
    "    for i in range(n):\n",
    "        u[i+1] = u[i] + f(t[i], u[i])*h\n",
    "    return (t, u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "Show that for the integration case $f(t, u) = f(t)$, Euler's method is the same as the composite left-hand endpoint rule,\n",
    "as in the section [Definite Integrals, Part 2](definite-integrals-2-composite-rules.ipynb)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving for [Example 1](#example-1): an integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_1(t, u):\n",
    "    \"\"\"For integration of -sin(x): should get cos(x) + constant\"\"\"\n",
    "    return -np.sin(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 0\n",
    "b = 2*np.pi\n",
    "u_0 = 1.\n",
    "n_steps = 50  # Often enough for a nice graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(x, Y) = euler(f_1, a, b, u_0, n=n_steps)  \n",
    "y = np.cos(x)\n",
    "figure(figsize=[12,8])\n",
    "title(\"This should look like $y = \\cos(x)$\")\n",
    "plot(x, y, 'g', label=\"Exact solution\")\n",
    "plot(x, Y, 'b:', label=\"Euler's answer\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving for [Example 2](#example-2): some exponential functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_2(t, u):\n",
    "    \"\"\"For solving du/dt = k u.\n",
    "    The variable k may be defined later, so long as that is done before this function is used.\n",
    "    \"\"\"\n",
    "    return k*u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# You could experiment by changing these values here;\n",
    "# for now I instead redefine them below.\n",
    "k = 1\n",
    "u_0 = 0.8\n",
    "a = 0\n",
    "b = 2\n",
    "n_steps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_2, a, b, u_0, n=n_steps)\n",
    "figure(figsize=[12,8])\n",
    "title(f\"This should look like $u = {u_0} \\, \\exp({k} \\, t)$\")\n",
    "#u = u_0*np.exp(k*t)\n",
    "u = np.exp(k*t)\n",
    "plot(t, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=\"Euler's answer\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# You could experiment by changing these values here.\n",
    "k = -0.5\n",
    "u_0 = 3\n",
    "a = 0\n",
    "b = 2\n",
    "n_steps = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_2, a, b, u_0, n=n_steps)\n",
    "figure(figsize=[12,8])\n",
    "title(f\"This should look like $y = {u_0} \\, \\exp({k} \\, t)$\")\n",
    "#u = u_0*np.exp(k*t)\n",
    "u = 4*np.exp(k*t)\n",
    "plot(t, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=\"Euler's answer\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving for [Example 3](#example-3); solutions that blow up"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_3(t, u):\n",
    "    \"\"\"For solving du/dt = u^2.\"\"\"\n",
    "    return u**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 0\n",
    "b = 0.9\n",
    "u_0 = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_3, a, b, u_0, n=50)\n",
    "u = 1/(a + 1/u_0 - t)\n",
    "figure(figsize=[12,8])\n",
    "title(f\"This should look like $u = 1/({a + 1/u_0} - t)$\")\n",
    "plot(t, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=\"Euler's answer\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is clearly a problem when $t$ reaches 1; let us explore that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = 1.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_3, a, b, u_0, n=50)\n",
    "T = a + 1/u_0\n",
    "t1000 = linspace(a, b, 1000)  # More t values are needed to get a good graph near the vertical asymptote\n",
    "u = 1/(a + 1/u_0 - t1000)\n",
    "\n",
    "figure(figsize=[12,8])\n",
    "title(f\"This should look like $u = 1/({T} - t)$\")\n",
    "plt.ylim(bottom=-100, top=1000)\n",
    "# Note: if you are curious about this new function \"ylim\", uncomment the following:\n",
    "#help(plt.ylim)\n",
    "plot(t1000, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=\"Euler's answer\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Clearly Euler's method can never produce the vertical asymptote; even less can it get the jump to negative values of $u$.\n",
    "The best we can do is improve accuracy by using more, smaller time steps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "b=0.999\n",
    "(t, U) = euler(f_3, a, b, u_0=1., n=10000)  # More t values, to improve accuracy\n",
    "T = a + 1/u_0\n",
    "t1000 = linspace(a, b, 1000)  # More t values are needed to get a good graph near the vertical asymptote\n",
    "u = 1/(a + 1/u_0 - t1000)\n",
    "\n",
    "figure(figsize=[12,8])\n",
    "title(f\"This should look like $u = 1/({T} - t)$\")\n",
    "plt.ylim(bottom=-100, top=1000)\n",
    "plot(t1000, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=\"Euler's answer\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name=\"solving-example-4\"></a>\n",
    "### Solving for [Example 4](#example-4) (Added March 22)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_4(t, u):\n",
    "    \"\"\"The variable K may be defined later, so long as that is done before this function is used.\n",
    "    The solution for a=0 is u(t) = u(t; 0, u_0) = \\cos t + (u_0 - 1) e^{-K t}\n",
    "    \"\"\"\n",
    "    return -np.sin(t) - K*(u - np.cos(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With enough steps (small enough step size $h$), all is well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 0.\n",
    "b = 2 * np.pi  # One period\n",
    "u_0 = 2.\n",
    "K = 40.\n",
    "n=1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_4, a, b, u_0, n)\n",
    "u = np.cos(t) + (u_0 - 1) * np.exp(-K*t) \n",
    "figure(figsize=[12,8])\n",
    "h = (b-a)/n\n",
    "title(f\"This should look like u = cos t + {u_0-1:0.3} exp(-{K} t)\")\n",
    "plot(t, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=f\"Euler's answer with h={h:0.3}\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, with large steps (still small enough to handle the $\\cos t$ part), there is a catastrophic failure, with growing oscillations that, as we will see, are a characteristic feature of *instability*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_4, a, b, u_0, n)\n",
    "u = np.cos(t) + (u_0 - 1) * np.exp(-K*t) \n",
    "figure(figsize=[12,8])\n",
    "h = (b-a)/n\n",
    "title(f\"This should look like u = cos t + {u_0-1:0.3} exp(-{K} t)\")\n",
    "plot(t, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=f\"Euler's answer with h={h:0.3}\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To show that the $K$ part is the problem, reduce $K$ while leaving the rest unchanged:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "(t, U) = euler(f_4, a, b, u_0, n)\n",
    "u = np.cos(t) + (u_0 - 1) * np.exp(-K*t) \n",
    "figure(figsize=[12,8])\n",
    "h = (b-a)/n\n",
    "title(f\"This should look like u = cos t + {u_0-1:0.3} exp(-{K} t)\")\n",
    "plot(t, u, 'g', label=\"Exact solution\")\n",
    "plot(t, U, 'b:', label=f\"Euler's answer with h={h:0.3}\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Variable Time Step Sizes $h_i$ (just a preview for now)\n",
    "\n",
    "It is sometime useful to adjust the time step size; for example reducing it when the derivative is larger,\n",
    "(as happens in Example 3 above).\n",
    "This gives a slight variant, now expressed in pseudo-code:\n",
    "\n",
    "Input: $f$, $a$, $b$, $n$ <br>\n",
    "$t_0 = a$ <br>\n",
    "$U_0 = u_0$ <br>\n",
    "for i in $[0, n)$: <br>\n",
    "$\\quad$ Choose $h_i$ somehow <br>\n",
    "$\\quad t_{i+1} = t_i + h_i$ <br>\n",
    "$\\quad U_{i+1} = U_i + h_i f(t_i, U_i)$ <br>\n",
    "end for\n",
    "\n",
    "In a later section, we will see how to estimate errors within an algorithm,\n",
    "and then how to use such error estimates to guide the choice of step size."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Error Analysis for the Canonical Test Case, $u' = k u$.\n",
    "\n",
    "A great amount of intuition about numerical methods for solving ODE IVPs comes from that \"simplest nontrivial example\", number 2 above.\n",
    "We can solve it with constant step size $h$, and thus study its errors and accuracy.\n",
    "The recursion relation is now\n",
    "\n",
    "$$U_{i+1} = U_i + h k U_i = U_i (1 + hk),$$\n",
    "\n",
    "with solution\n",
    "\n",
    "$$U_i = u_0 (1 + hk)^i.$$\n",
    "\n",
    "For comparison, the exact solution of this ODE IVP is\n",
    "\n",
    "$$\n",
    "u(t_i) = u_0 e^{k(t_i - a)} = u_0 e^{k ih} = u_0 (e^{kh})^i\n",
    "$$\n",
    "\n",
    "So each is a geometric series: the difference is that the *growth factor* is $G = (1 + hk)$ for Euler's method,\n",
    "vs $g = e^{kh} = 1 + hk + (hk)^2/2 + \\cdots = 1 + hk + O(h^2)$ for the ODE.\n",
    "\n",
    "Ths deviation at each time step is $O(h^2)$, suggesting that by the end $t=b$, at step $n$, the error will be $O(n h^2) = O\\left(\\displaystyle\\frac{b-a}{h} h^2\\right) = O(h)$.\n",
    "\n",
    "This is in fact what happens, but to verify that, we must deal with the challenge that once an error enters at one step, it is potentially amplified at each subsequent step, so the errors introduced at each step do not simply get summed like they did with definite integrals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Global Error and Local (Truncation) Error\n",
    "\n",
    "Ultimately, the error we need to understand is the **global error**: at step $i$,\n",
    "\n",
    "$$ E_i = u(t_i) - U_i $$\n",
    "\n",
    "We will approach this by first considering the new error added at each step, the **local truncation error** (or **discretization error**).\n",
    "\n",
    "At the first step this is the same as above:\n",
    "\n",
    "$$ e_1 = u(t_1) - U_1 = u(a+h) - U_1 $$\n",
    "\n",
    "However at later steps we compare the results $U_{i+1}$ to what the solution would be if it were exact at the start of that step: that is, if $U_i$ were exact.\n",
    "\n",
    "Using the notation $u(t; t_i, U_i)$ introduced <a href=\"#IVP_notation\">above</a> for the solution of the ODE with initial condition $u(t_i) = U_i$, the **location truncation error** at step $i$ is the discrepancy at time $t_{i+1}$ between what Euler's method and the exact solution give when both start at that point $(t_i, U_i)$:\n",
    "\n",
    "$$ e_i = u(t; t_i, U_i) - U_{i+1} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Error propagation in $u' = k u$, $k \\geq 0$.\n",
    "\n",
    "After one step, $E_1 = u(t_1) - U_1 = e_1$.\n",
    "\n",
    "At step 2,\n",
    "\n",
    "$$ E_2 = u(t_2) - U_2 = (u(t_2) - u(t_2, t_1, U_1) + (u(t_2, t_1, U_1) - U_2)\n",
    "= (u(t_2) - u(t_2, t_1, U_1) + e_2 $$\n",
    "\n",
    "The first term is the difference at $t=t_2$ of two solutions with values at $t = t_1$ being $u(t_1)$ and $U_1$ respectively. As the ODE is linear and homogeneous, this is the solution of the same ODE with value at $t=t_1$ being $u(t_1) - U_1$, which is $e_1$:\n",
    "that solution is $e_1 e^{y(t - t_1}$, so at $t=t_2$ it is $e_1 e^{kh}$.\n",
    "Thus the global error after two steps is\n",
    "\n",
    "$$ E_2 = e_2 + (e^{kh})e_1: $$\n",
    "\n",
    "the error from the previous step has been amplified by the growth factor $g = e^{kh}$:\n",
    "\n",
    "$$ E_2 = e_2 + g e_1: $$\n",
    "\n",
    "This continues, so that\n",
    "\n",
    "$$ E_3 = e_3 + g E_1 = e_3 + g (e_2 + g e_1) = e_3 + ge_2 + g^2 e_1 $$\n",
    "\n",
    "and so on, leading to\n",
    "\n",
    "$$ E_i = e_i + g e_{i-1} + g^2 e_{i-2} + \\cdots + g^{i-1} e_{1} $$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bounding the local truncation errors ...\n",
    "\n",
    "To get a bound on the global error from the formula above, we first need a bound on the local truncation errors $e_i$.\n",
    "\n",
    "Taylor's theorem gives $e^{kh} = 1 + kh + e^{k \\xi} (kh)^2/2$, $0 < \\xi < k h$, so\n",
    "\n",
    "$$\n",
    "e_i = U_i e^{kh} - U_i (1 + kh) = U_i (e^{k \\xi} h^2/2)\n",
    "$$\n",
    "\n",
    "and thus\n",
    "\n",
    "$$|e_i| \\leq |U_i| \\frac{e^{kh}}{2} h^2$$\n",
    "\n",
    "Also, since $1 + kh < e^{kh}$,\n",
    "$|U_i| < |u(t_i)| = |u_0| e^{k(t_i - a)},$\n",
    "and we only need this to the beginning of the last step, $i \\leq n-1$, for which\n",
    "\n",
    "$$\n",
    "|U_i| < |u_0| e^{k(b - h - a)}\n",
    "$$\n",
    "\n",
    "Thus\n",
    "\n",
    "$$\n",
    "|e_i| \\leq \\frac{|u_0| e^{k(b - h - a)} e^{kh}}{2} h^2 = \\frac{|u_0 e^{k(b - a)}|}{2} h^2\n",
    "$$\n",
    "\n",
    "That is,\n",
    "\n",
    "$$ |e_i| \\leq C h^2 \\text{ where } C := \\frac{|u_0 e^{k(b - a)}|}{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ... and using this to complete the bound on the global truncation error\n",
    "\n",
    "Using this bound on the local errors $e_i$ in the above sum for the global error $E_i$,\n",
    "\n",
    "$$ | E_i | \\leq C h^2 (1 + g + \\cdots g^{i-1}) = C \\frac{g^i - 1}{g-1} h^2 $$\n",
    "\n",
    "Since $g^i = e^{k h i} = e^{k (t_i - a)}$ and the denominator $g - 1 = e^{kh} - 1 > k h$,\n",
    "we get\n",
    "\n",
    "$$\n",
    "| E_i | \\leq  C \\frac{e^{k (t_i - a)} - 1}{k h} h^2\n",
    "\\leq \\frac{|u_0 e^{k(b - a)}|}{2} \\frac{e^{k (t_i - a)} - 1}{k} h, = O(h)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this global error formula is bulilt from three factors:\n",
    "\n",
    "- The first is the constant\n",
    "$\\displaystyle \\frac{|u_0 e^{k(b - a)}|}{2}$\n",
    "which is roughly half of the maximum value of the exact solution over the interval $[a, b]$.\n",
    "\n",
    "- The second\n",
    "$\\displaystyle \\frac{e^{k (t_i - a)} - 1}{k}$\n",
    "depends on $t$, and \n",
    "\n",
    "- The third is $h$, showing the overall order of accuracy: first order: the overall absolute error is $O(h)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A more general error bound\n",
    "\n",
    "A very similar result applies to the solution $u(t; a, u_0)$ of the more general initial value problem\n",
    "\n",
    "$$\n",
    "\\frac{du}{dt} = f(t, u), \\quad u(a) = u_0\n",
    "$$\n",
    "\n",
    "so long as the function $f$ is \"somewhat well-behaved\" in that it satisfies a so-called *Lipschitz Condition*:\n",
    "that there is some constant $K$ such that\n",
    "\n",
    "$$\n",
    "\\left| \\frac{\\partial F}{\\partial u}(t, u) \\right| \\leq K \n",
    "$$\n",
    "\n",
    "for the relevant time values $a \\leq t \\leq b$.\n",
    "\n",
    "(**Aside:** As you might have seen in a course on differential equations, such a Lipschitz condition is necessary to even guarantee that the initial value problem has a unique solution, so it is a quite reasonable requirement.)\n",
    "\n",
    "Then this constant $K$ plays the part of the exponential growth factor $k$ above:\n",
    "\n",
    "first one shows that the local trunction error is bounded by\n",
    "\n",
    "$$ |e_i| \\leq C h^2 \\text{ where now } C := \\frac{|u_0 e^{K(b - a)}|}{2}; $$\n",
    "\n",
    "then calculating as above bounds the global truncation error with\n",
    "\n",
    "$$\n",
    "| E_i | \\leq \\frac{|u_0 e^{K(b - a)}|}{2} \\frac{e^{K (t_i - a)} - 1}{k} h, = O(h)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### There is much room for improvement\n",
    "\n",
    "As with definite integrals, this is not very impressive, so in the next section on\n",
    "[Runge-Kutta Methods](ODE-IVP-2-Runge-Kutta-python.ipynb)\n",
    "we will explore several widely used methods that improve to second order and then fourth order accuracy.\n",
    "Later, we will see how to get even higher orders."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But first, we can illustrate how this exponential growth of errors looks in some examples, and coapr the the better behaved errors in definite integrals.\n",
    "\n",
    "This will be done by looking at the effect of a small change in the initial value, to simulate an error that arises there."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Error propagation for [Example 1](#example-1), an integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 0\n",
    "b = 2*np.pi\n",
    "u_0 = 1  # Original value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "(x, u) = euler(f_1, a, b, u_0)  # Use the default of 50 steps; typically enough for a nice graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But now \"perturb\" the initial value in all cases by this much:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "delta_u_0 = 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "(x, u_perturbed) = euler(f_1, a, b, u_0+delta_u_0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=[12,8])\n",
    "title(\"The solution before perturbing $u(0)$ was $u = \\cos(x)$\")\n",
    "plot(x, np.cos(x), 'g', label=\"Original exact solution\")\n",
    "plot(x, u, 'b:', label=\"Euler's answer before perturbation\")\n",
    "plot(x, u_perturbed, 'r:', label=\"Euler's answer after perturbation\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This just shifts all the $u$ values up by the perturbation of $u_0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Error propagation for [Example 2](#example-2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = 1\n",
    "a = 0\n",
    "b = 2\n",
    "u_0 = 1  # Original value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "(t, U) = euler(f_2, a, b, u_0)\n",
    "(t, U_perturbed) = euler(f_2, a, b, u_0+delta_u_0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=[12,8])\n",
    "title(\"The solution before perturbing $u(0)$ was $u = {u_0} \\, \\exp({k} \\, t)$\")\n",
    "plot(t, u_0*np.exp(k*t), 'g', label=\"Original exact solution\")\n",
    "plot(t, U, 'b:', label=\"Euler's answer before perturbation\")\n",
    "plot(t, U_perturbed, 'r:', label=\"Euler's answer after perturbation\")\n",
    "legend()\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Graphing the error shows its exponential growth:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=[12,8])\n",
    "title(\"Error\")\n",
    "plot(t, u_0*np.exp(k*t) - U)\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "This work is licensed under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/)"
   ]
  }
 ],
 "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
