09_ODE_ivp_part1.ipynb

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    "<table>\n",
    " <tr align=left><td><img align=left src=\"./images/CC-BY.png\">\n",
    " <td>Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Kyle T. Mandli</td>\n",
    "</table>\n",
    "\n",
    "Note:  The presentation below largely follows part II in \"Finite Difference Methods for Ordinary and Partial Differential Equations\" by LeVeque (SIAM, 2007)."
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   "source": [
    "from __future__ import print_function\n",
    "\n",
    "%matplotlib inline\n",
    "import numpy\n",
    "import matplotlib.pyplot as plt"
   ]
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   "source": [
    "# Numerical Solution to ODE Initial Value Problems - Part 1"
   ]
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   "source": [
    "Many physical, biological, and societal systems can be written as a system of ordinary differential equations (ODEs).  In the case where the initial state (value) is know the problems can be written as\n",
    "\n",
    "$$\n",
    "    \\frac{\\text{d}\\mathbf{u}}{\\text{d}t} = \\mathbf{f}(t, \\mathbf{u}) \\quad \\mathbf{u}(0) = \\mathbf{u}_0\n",
    "$$\n",
    "\n",
    "where\n",
    " - $\\mathbf{\\!u}(t)$ is the state vector\n",
    " - $\\mathbf{\\!f}(t, \\mathbf{\\!u})$ is a vector-valued function that describes the evolution of $\\mathbf{u}$ with time\n",
    " - $\\mathbf{\\!u}(0)$ is the initial condition at time $t = 0$"
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    "### The Example for our time:  A non-linear  model of Epidemics\n",
    "\n",
    "Classical [Kermack and McKendrick (1927)](https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0118) SIR model of epidemics (with reinfection)\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\frac{ds}{dt} &= -si + kr \\\\\n",
    "    \\frac{di}{dt} &= si -\\sigma i \\\\\n",
    "    \\frac{dr}{dt} &= \\sigma i - kr\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Where the variable $s$ represents the fraction of a population that is **Susceptible** to infection, $i$ is the proportion **Infected** and $r$ the fraction **Recovered**.  For this model $s+i+r =1$.  The parameters $\\sigma, k \\geq 0$ control the relative rates of infection and recovery.  "
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    "For this problem\n",
    "$$\n",
    "\\mathbf{u}(t) = \\begin{bmatrix} s(t)\\\\ i(t)\\\\ r(t)\\\\\\end{bmatrix},\\quad\\quad\n",
    "\\mathbf{f}(t,\\mathbf{u}) = \\begin{bmatrix} -si + kr \\\\ si -\\sigma i \\\\ \\sigma i - kr\\\\\\end{bmatrix}\n",
    "$$"
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   "source": [
    "### Numerical Solutions "
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    "# Solve using SciPy's ODE integrator solve_ivp\n",
    "from scipy.integrate import solve_ivp\n",
    "\n",
    "# define the RHS of our system of ODE's\n",
    "def f_sir(t, u, sigma, k):\n",
    "    s,i,r = u\n",
    "    return numpy.array([-s*i + k*r,\n",
    "                        (s - sigma)*i,\n",
    "                        sigma*i - k*r ])"
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    "sigma = .2\n",
    "k = 0.01\n",
    "t_max = 100\n",
    "u_0 = [0.999, 0.001, 0.]\n",
    "sol = solve_ivp(f_sir, [0, t_max] , u_0, args=(sigma, k), rtol=1.e-6, atol=1.e-9,dense_output = True)"
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    "t = numpy.linspace(0, t_max, 300)\n",
    "z = sol.sol(t)\n",
    "\n",
    "fig = plt.figure(figsize=(20,7))\n",
    "axes = fig.add_subplot(1,2,1)\n",
    "axes.plot(t,z[0],'r',label='s', linewidth=2)\n",
    "axes.plot(t,z[1],'b',label='i', linewidth=2)\n",
    "axes.plot(t,z[2],'g',label='r',  linewidth=2)\n",
    "axes.plot(t,sigma*numpy.ones(t.shape),'k--',label='$\\sigma$')\n",
    "\n",
    "axes.legend(loc='best',shadow=True, fontsize=14)\n",
    "axes.set_xlabel('Time',fontsize=16)\n",
    "axes.set_ylabel('Population',fontsize=16)\n",
    "axes.grid()\n",
    "axes.set_title('SIR system: $\\sigma={}$, $k={}$'.format(sigma,k),fontsize=18)\n",
    "plt.show()"
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    "### Questions an epidemiologist might ask:\n",
    " - What are the dynamics of this system?  does it predict steady or oscillatory solutions?\n",
    " - Can we estimate critical parameters ($\\sigma$, $k$) from data?\n",
    " - Can we reliably use this model to predict the future?\n",
    " - How do we evaluate whether this is a *useful* model?\n",
    " - How might I modify/improve this model.\n",
    " "
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    " \n",
    "### Questions a Computational Mathematician might ask:\n",
    " - Does a solution to the model even exist and is it unique?\n",
    " - Is our approximate numerical solution accurate?\n",
    " - What are the dynamics of this system?  does it predict steady or oscillatory solutions?\n",
    " - how do we understand the sensitivity to parameters?"
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    "### Existence and Uniqueness of solutions (n-D autonomous systems)\n",
    "\n",
    "For proof see [Hirsch, Smale, Devaney, Dynamical Systems](https://www.amazon.com/Differential-Equations-Dynamical-Systems-Introduction/dp/0123820103)"
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    "#### Theorem: (Picard-Lindelhof)\n",
    "\n",
    "Given an Autonomous, dynamical system\n",
    "\n",
    "$$\n",
    "    \\frac{\\text{d}\\mathbf{u}}{\\text{d}t} = \\mathbf{f}(\\mathbf{u}) \\quad \\mathbf{u}(0) = \\mathbf{u}_0\n",
    "$$\n",
    "\n",
    "with $\\mathbf{u}\\in\\mathbb{R}^n$ and $\\mathbf{f}:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n$\n",
    "\n",
    "Consider a \"spherical\" domain of radius $\\rho$ around the initial condition $\\mathbf{u}_0$.  \n",
    "\n",
    "If, within this domain,  $\\mathbf{f}$ is \n",
    "\n",
    "* Bounded: $|\\mathbf{f}| < M$\n",
    "* Lipshitz Continuous: $$|\\mathbf{f}(\\mathbf{x}) - \\mathbf{f}(\\mathbf{y})| < K|\\mathbf{x}- \\mathbf{y} |$$\n",
    "\n",
    "Then a unique solution exists to the ODE IVP for some interval of time $t\\in[-a,a]$ where $0 < a <  \\min(\\rho/M, 1/K)$"
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   "source": [
    "### Geometric Picture\n",
    "\n",
    "<table>\n",
    "    <tr align=center><td><img align=left src=\"./images/Picard-Lindelhof-figure.png\" width=800></td>\n",
    "        <td>$$\n",
    "    \\frac{\\text{d}\\mathbf{u}}{\\text{d}t} = \\mathbf{f}(\\mathbf{u}), \\quad \\mathbf{u}(0) = \\mathbf{u}_0\n",
    "$$</td>\n",
    "</table>"
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   "source": [
    "* **Short Version**: If $\\mathbf{f}$ is sufficiently smooth, then a local solution to the ODE exists and is unique"
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    "* **Caveat**: The theorem itself gives *NO* constructive way to find that solution"
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    "#### Other Examples:  Simple radioactive decay\n",
    "$$\n",
    "    \\mathbf{\\!u} = [c], \\quad \\mathbf{f} = [-\\lambda c]\n",
    "$$\n",
    "   \n",
    "$$\n",
    "    \\frac{\\text{d} c}{\\text{d}t} = -\\lambda c \\quad c(0) = c_0\n",
    "$$\n",
    "   \n",
    "\n",
    "which has solutions of the form $c(t) = c_0 e^{-\\lambda t}$"
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    "decay_constant = -numpy.log(2.)/1600.\n",
    "t = numpy.linspace(0., 4800, 11)\n",
    "y = numpy.linspace(0., 1.2, 11)\n",
    "T, Y = numpy.meshgrid(t,y)\n",
    "dt = numpy.ones(T.shape)\n",
    "dy = -dt*Y\n",
    "\n",
    "tp = numpy.linspace(0., 4800, 100)\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.quiver(T,Y, dt,dy,linewidth=0.1,color='gray')\n",
    "axes.plot(tp,numpy.exp(decay_constant*tp),linewidth=3)\n",
    "axes.plot(0.,1.,'ro', markersize=10)\n",
    "axes.plot(1600., 0.5,'rx',markersize=10)\n",
    "axes.grid()\n",
    "axes.set_title(\"Radioactive Decay, $u' = - \\lambda u$, $u(0)=1$, $t_{1/2}=1600$ yr\", fontsize=18)\n",
    "axes.set_xlabel('t (years)', fontsize=16)\n",
    "axes.set_ylabel('u', fontsize=16)\n",
    "\n",
    "axes.set_ylim((-.1,1.2))\n",
    "plt.show()\n",
    "\n"
   ]
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    "#### Examples:  Complex radioactive decay (or chemical system).\n",
    "\n",
    "Chain of decays from one species to another.\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    \\frac{\\text{d} c_1}{\\text{d}t} &= -\\lambda_1 c_1 \\\\\n",
    "    \\frac{\\text{d} c_2}{\\text{d}t} &= \\lambda_1 c_1 - \\lambda_2 c_2 \\\\\n",
    "    \\frac{\\text{d} c_3}{\\text{d}t} &= \\lambda_2 c_2 - \\lambda_3 c_3 \n",
    "\\end{aligned}$$"
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   "source": [
    "$$\\frac{\\text{d} \\mathbf{u}}{\\text{d}t} = \\frac{\\text{d}}{\\text{d}t}\\begin{bmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{bmatrix} = \n",
    "\\begin{bmatrix} \n",
    "    -\\lambda_1 & 0 & 0 \\\\\n",
    "    \\lambda_1 & -\\lambda_2 & 0 \\\\\n",
    "    0 & \\lambda_2 & -\\lambda_3\n",
    "\\end{bmatrix} \\begin{bmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{bmatrix}$$\n",
    "\n",
    "$$\\frac{\\text{d} \\mathbf{u}}{\\text{d}t} = A \\mathbf{u}$$\n",
    "\n",
    "For systems of equations like this the general solution to the ODE is the matrix exponential:\n",
    "\n",
    "$$\\mathbf{u}(t) = e^{A t}\\mathbf{u}_0 = \\sum_{i=1}^3 c_i\\mathbf{s}_i e^{-\\lambda_i t}$$\n",
    "\n",
    "which can be solved given the eigenvalues and eigenvectors of $A$."
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   "source": [
    "#### Examples:  Particle tracking in a fluid\n",
    "\n",
    "$$\\frac{\\text{d} \\mathbf{X}}{\\text{d}t} = \\mathbf{V}(t, \\mathbf{X})$$\n",
    "\n",
    "In fact all ODE IVP systems can be thought of as tracking particles through a flow field (dynamical system).  In 1-dimension the flow \"manifold\" we are on is fixed by the initial condition."
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    "x = numpy.linspace(0., 1., 11)\n",
    "y = numpy.linspace(0., 1., 11)\n",
    "x_fine = numpy.linspace(0., 1.)\n",
    "y_fine = numpy.linspace(0., 1.)\n",
    "\n",
    "X, Y = numpy.meshgrid(x,y)\n",
    "X_fine, Y_fine = numpy.meshgrid(x_fine, y_fine)\n",
    "\n",
    "pi = numpy.pi\n",
    "psi = numpy.sin(pi*X_fine)*numpy.sin(pi*Y_fine)\n",
    "U = pi*numpy.sin(pi*X)*numpy.cos(pi*Y)\n",
    "V = -pi*numpy.cos(pi*X)*numpy.sin(pi*Y)\n",
    "\n",
    "x0 = 0.75\n",
    "y0 = 0.75\n",
    "psi0 = numpy.sin(pi*x0)*numpy.sin(pi*y0)\n",
    "\n",
    "fig = plt.figure(figsize=(8,8))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.quiver(X,Y, U, V)\n",
    "axes.plot(.75, 0.75,'ro')\n",
    "axes.contour(X_fine, Y_fine, psi, [ psi0 ])\n",
    "axes.grid()\n",
    "axes.set_title(\"Particle tracking\", fontsize=18)\n",
    "axes.set_xlabel('y', fontsize=16)\n",
    "axes.set_ylabel('x', fontsize=16)\n",
    "\n",
    "plt.show()"
   ]
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   "source": [
    "#### Examples: Van der Pol Oscillator\n",
    "\n",
    "$$y'' - \\mu (1 - y^2) y' + y = 0 \\quad \\quad \\text{with} \\quad \\quad  y(0) = y_0, \\quad y'(0) = v_0$$\n",
    " "
   ]
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   "source": [
    "$$\\mathbf{u} = \\begin{bmatrix} y \\\\ y' \\end{bmatrix} = \\begin{bmatrix} u_1 \\\\ u_2 \\end{bmatrix}$$\n",
    "   \n",
    "$$\\frac{\\text{d}}{\\text{d}t} \\begin{bmatrix} u_1 \\\\ u_2 \\end{bmatrix} = \\begin{bmatrix} u_2 \\\\ \\mu (1 - u_1^2) u_2 - u_1 \\end{bmatrix} = \\mathbf{f}(t, \\mathbf{u})$$"
   ]
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   "source": [
    "from scipy.integrate import solve_ivp\n",
    "\n",
    "def f_vanderpol(t, u, mu=5.):\n",
    "    return numpy.array([u[1], mu * (1.0 - u[0]**2) * u[1] - u[0]])\n",
    "\n",
    "# N = 100\n",
    "N = 500\n",
    "t_span = (0., 200.)\n",
    "u0 = [ 1., 10. ]\n",
    "f = lambda t, u: f_vanderpol(t, u, mu=50.)\n",
    "sol = solve_ivp(f, t_span, u0,method='BDF')"
   ]
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    "fig = plt.figure(figsize=(16,6))\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "\n",
    "axes.plot(sol.t, sol.y[0])\n",
    "axes.set_title(\"Solution to Van der Pol Oscillator\", fontsize=18)\n",
    "axes.set_xlabel(\"t\", fontsize=16)\n",
    "axes.set_ylabel(\"y(t)\", fontsize=16)\n",
    "axes.grid()\n",
    "\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "\n",
    "axes.plot(sol.y[0],sol.y[1],'r')\n",
    "axes.set_title(\"Phase Diagram for Van der Pol Oscillator\", fontsize=18)\n",
    "axes.set_xlabel(\"y(t)\", fontsize=16)\n",
    "axes.set_ylabel(\"y'(t)\", fontsize=16)\n",
    "axes.grid()\n",
    "plt.show()"
   ]
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    "## The Big Questions\n",
    "\n",
    "Given a RHS $\\mathbf{f}(t,\\mathbf{u})$ and initial condition $\\mathbf{u}(t_0) \\ldots$\n",
    "\n",
    "*  How do you find a discrete numerical solution that approximates the trajectory $\\mathbf{u}(t)$?\n",
    "*  How do you control the accuracy of the approximation?\n",
    "*  How do you improve the efficiency of the approximation?\n",
    "*  How do you understand  stability and convergence?"
   ]
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    "## Some Notation:  Basic Stepping schemes\n",
    "\n",
    "Introducing some notation to simplify things\n",
    "$$\\begin{aligned}\n",
    "    t_0 &= 0 \\\\\n",
    "    t_1 &= t_0 + \\Delta t \\\\\n",
    "    t_n &= t_{n-1} + \\Delta t = n \\Delta t + t_0 \\\\\n",
    "    u_0 &= u(t_0) \\approx U_0 \\\\\n",
    "    u_1 &= u(t_1) \\approx U_1 \\\\\n",
    "    u_n &= u(t_n) \\approx U_2 \\\\\n",
    "\\end{aligned}$$\n",
    "where lower-case letters are \"exact\".  "
   ]
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   "source": [
    "Looking back at our work on numerical differentiation why not approximate the derivative as a finite difference:\n",
    "\n",
    "$$\n",
    "    \\frac{u(t + \\Delta t) - u(t)}{\\Delta t} = f(t, u)\n",
    "$$\n",
    "\n",
    "We still need to decide how to evaluate the $f(t, u)$ term however.  "
   ]
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   "source": [
    "One obvious way to do this, is to just use $f(t, u(t))$ and write the update scheme as\n",
    "\n",
    "$$\n",
    "    u(t + \\Delta t) = u(t) + \\Delta t f(t,u(t))\n",
    "$$\n",
    "\n",
    "Which is our first integration scheme (which goes by the name Euler's method).  As usual, though the first scheme is often the worst scheme, but with a bit of understanding we can do much better with not a lot more work."
   ]
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    "### Integral form of ODE IVP's: the  Relationship to quadrature\n",
    "\n",
    "Euler's method is an example of a \"Single Step, multi-stage\" scheme of which there are many.  However, to derive them it is actually more instructive to work with the integral form of an ODE, which will put the equations in a form where we can use our ideas from quadrature to make progress"
   ]
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   "source": [
    "Given a a system of ODE's\n",
    "$$\n",
    "    \\frac{d\\mathbf{u}}{dt} = \\mathbf{f}(t,\\mathbf{u})\n",
    "$$"
   ]
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    "We can integrate both sides\n",
    "$$\n",
    "    \\int^{t + \\Delta t}_t \\frac{d\\mathbf{u}}{d \\tau} d\\tau = \\int^{t + \\Delta t}_t \\mathbf{f}(\\tau, \\mathbf{u}) d\\tau\n",
    "$$\n",
    "\n",
    "which is equivalent to the differential form. "
   ]
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   },
   "source": [
    " However using the fundamental theorem of calculus tells us that the LHS is $u(t + \\Delta t) - u(t)$  so we can write the ODE as\n",
    "\n",
    "$$ \n",
    "   \\mathbf{u}(t + \\Delta t) = \\mathbf{u}(t) + \\int^{t + \\Delta t}_t \\mathbf{f}(\\tau, \\mathbf{u}(\\tau)) d\\tau\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Single-Step Multi-Stage Schemes\n",
    "\n",
    "The integral form of an ODE initial value problem can be written\n",
    "\n",
    "$$ \n",
    "    u(t + \\Delta t) = u(t) + \\int^{t + \\Delta t}_t f(\\tau, u(\\tau)) d\\tau\n",
    "$$\n",
    "\n",
    "Which says that our solution $u$, if it exists  at some time $\\Delta t$ in the future,  is $u(t)$ plus a *number* \n",
    "\n",
    "$$\n",
    "    K = \\int^{t + \\Delta t}_t f(\\tau, u(\\tau) )d\\tau\n",
    "$$\n",
    "\n",
    "which is a definite *line integral* (along an unknown solution). \n",
    "\n",
    "An important class of ODE solvers are called *Single Step, Multi-stage schemes* which can be most easily understood as extensions of the  Newton-Cotes quadrature schemes for approximating $K$ (plus an error term that will scale as $\\Delta t^p$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The Geometric picture\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "t = numpy.linspace(0., 4800, 11)\n",
    "y = numpy.linspace(0., 1.2, 11)\n",
    "T, Y = numpy.meshgrid(t,y)\n",
    "dt = numpy.ones(T.shape)\n",
    "dy = -dt*Y\n",
    "\n",
    "tK = 2000.\n",
    "uK = numpy.exp(decay_constant*tK)\n",
    "K = uK -1.\n",
    "tp = numpy.linspace(0., 4800, 100)\n",
    "tk = numpy.linspace(0., tK, 100)\n",
    "\n",
    "fig = plt.figure(figsize=(10,8))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.quiver(T,Y, dt,dy, color='gray')\n",
    "axes.plot(tp,numpy.exp(decay_constant*tp))\n",
    "axes.plot(0.,1.,'ro')\n",
    "axes.plot(tk,numpy.exp(decay_constant*tk),'r--')\n",
    "axes.plot(tK, uK, 'ro')\n",
    "axes.plot([0.,0.], [1., uK], 'r--')\n",
    "axes.text(10., 0.72, '$K$', fontsize=24, color='red')\n",
    "axes.plot([0.,tK],[uK, uK], 'r--')\n",
    "axes.text(900., uK - .1, '$\\Delta t$', fontsize=24, color='red')\n",
    "axes.text(-10, 1., '$U_0$', fontsize=24, color='blue')\n",
    "axes.text(tK+10, uK, '$U_1$', fontsize=24, color='blue')\n",
    "\n",
    "\n",
    "axes.grid()\n",
    "axes.set_title(\"Direction Set, $u' = - \\lambda u$, $u(0)=1$\", fontsize=18)\n",
    "axes.set_xlabel('t (years)', fontsize=16)\n",
    "axes.set_ylabel('u', fontsize=16)\n",
    "\n",
    "axes.set_ylim((-.1,1.2))\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "#### Exercise (left for the reader)\n",
    "\n",
    "Show that \n",
    "$$\n",
    "    u(\\Delta t) = u_0 + \\int_0^{\\Delta t} f(\\tau, u(\\tau)) d\\tau\n",
    "$$\n",
    "\n",
    "when $f(\\tau, u(\\tau)) = -\\lambda u(\\tau)$ and $u(\\tau) = u_0e^{-\\lambda \\tau}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Forward Euler scheme\n",
    "For example, if we approximate $K$ with a left-sided quadrature rule\n",
    "$$\n",
    "    K = \\int^{t + \\Delta t}_t f(\\tau, u(\\tau)) d\\tau \\approx \\Delta t f(t, u(t))\n",
    "$$\n",
    "\n",
    "then our first ODE algorithm can be written\n",
    "$$\n",
    "u(t + \\Delta t) =  u(t) + \\Delta t f(t, u(t))\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Which is exactly Euler's method that we derived previously"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "in terms of our discrete approximation $U$\n",
    "$$\n",
    "\\begin{align}\n",
    "    K_1 &= \\Delta t f(t_n, U_n)\\\\\n",
    "    U_{n+1} &= U_n + K_1\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    " known as the *forward Euler method*.  In essence we are approximating the derivative with the value of the function at the point we are at $t_n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "t = numpy.linspace(0.0, 1.6e3, 100)\n",
    "c_0 = 1.0\n",
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "\n",
    "# Euler step\n",
    "dt = 1e3\n",
    "u_np = c_0 + dt * (decay_constant * c_0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, c_0 * numpy.exp(decay_constant * t), label=\"True Solution\")\n",
    "axes.plot(0., 1., 'ro')\n",
    "axes.text(0., 1.01, '$U_0$', fontsize=16)\n",
    "axes.plot((0.0, dt), (c_0, u_np), 'k')\n",
    "axes.plot((dt, dt), (u_np, c_0 * numpy.exp(decay_constant * dt)), 'k--')\n",
    "axes.plot((0.0, 0.0), (c_0, u_np), 'k--')\n",
    "axes.text(10., 0.75, '$K_1$', fontsize=16)\n",
    "axes.plot((0.0, dt), (u_np, u_np), 'k--')\n",
    "axes.text(400, u_np - 0.05, '$\\Delta t$', fontsize=16)\n",
    "\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.05))\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Implement Forward Euler\n",
    "def euler(f, t_span, u0, N):\n",
    "    \"\"\" simple implementation of constant step-size forward euler method\n",
    "        This doc string should have so much more in it\n",
    "    \"\"\"\n",
    "    t = numpy.linspace(t_span[0], t_span[1],N)\n",
    "    u = numpy.empty(t.shape)\n",
    "    u[0] = u0\n",
    "    delta_t = t[1] - t[0]\n",
    "    for (n, t_n) in enumerate(t[:-1]):\n",
    "        K1 = delta_t * f(t_n, u[n])\n",
    "        u[n + 1] = u[n] + K1        \n",
    "    return t, u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "\n",
    "t_span = [0.0, 1.6e3]\n",
    "u0 = 1.\n",
    "N = 1000\n",
    "t_euler, u_euler = euler(f, t_span, u0, N)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "t_exact = numpy.linspace(0.0, 1.6e3, 100)\n",
    "u_exact = lambda t : c_0 * numpy.exp(decay_constant * t)\n",
    "\n",
    "fig = plt.figure(figsize=(16, 6))\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "axes.plot(t_euler, u_euler, 'or', label=\"Euler\")\n",
    "axes.plot(t_exact, u_exact(t_exact), 'k--', label=\"True Solution\")\n",
    "\n",
    "axes.set_title(\"Forward Euler\")\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_ylabel(\"$c(t)$\")\n",
    "axes.set_ylim((0.4,1.1))\n",
    "axes.grid()\n",
    "axes.legend()\n",
    "\n",
    "abs_err = numpy.abs(u_euler - u_exact(t_euler))\n",
    "rel_err = abs_err/u_exact(t_euler)\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "axes.plot(t_euler,abs_err,'ro',label='absolute error')\n",
    "axes.plot(t_euler,rel_err,'bo',label='relative error')\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_ylabel(\"error\")\n",
    "axes.set_title('Error')\n",
    "axes.legend(loc='best')\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Backward's Euler\n",
    "\n",
    "Similar to forward Euler is the *backward Euler* method which,   uses a right-rectangle rule to estimate $K$ given $f$ at a future time. i.e. \n",
    "\n",
    "$$\n",
    "    K\\approx \\Delta t f(t_{n+1}, U_{n+1})\n",
    "$$\n",
    "\n",
    "However, the update scheme now becomes\n",
    "$$\n",
    "    U_{n+1} = U_n + \\Delta t f(t_{n+1}, U_{n+1}).\n",
    "$$\n",
    "\n",
    "which requires a (usually non-linear) solve for $U_{n+1}$. Schemes where the function $f$ is evaluated at the unknown time are called *implicit methods*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For some cases we can solve the equation by hand.  For instance in the case of our example problem, $f=\\lambda U$, we have:\n",
    "\n",
    "$$\n",
    "    U_{n+1} = U_n + \\Delta t f(t_{n+1}, U_{n+1}) = U_n + \\Delta t (\\lambda U_{n+1})\n",
    "$$\n",
    "\n",
    "which can be solved for $U_{n+1}$ to find\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    U_{n+1} &= U_n + \\Delta t (\\lambda U_{n+1}) \\\\\n",
    "    U_{n+1} \\left[ 1 - \\Delta t \\lambda \\right ] &= U_n \\\\\n",
    "    U_{n+1} &= \\frac{U_n}{1 - \\Delta t \\lambda}\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "t = numpy.linspace(0.0, 1.6e3, 100)\n",
    "c_0 = 1.0\n",
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, c_0 * numpy.exp(decay_constant * t), label=\"True Solution\")\n",
    "\n",
    "# Plot Backwards Euler step\n",
    "dt = 1e3\n",
    "u_np = c_0 + dt * (decay_constant * c_0 * numpy.exp(decay_constant * dt))\n",
    "axes.plot((0.0, dt), (c_0, u_np), 'k')\n",
    "axes.plot(dt, u_np, 'ro')\n",
    "axes.text(dt+ 10., u_np, '$U_1$', fontsize=16)\n",
    "axes.plot((0.0, 0.0), (c_0, u_np), 'k--')\n",
    "axes.plot((0.0, dt), (u_np, u_np), 'k--')\n",
    "axes.text(400, u_np - 0.05, '$\\Delta t$', fontsize=16)\n",
    "axes.text(10., 0.85, '$K_1$', fontsize=16)\n",
    "\n",
    "axes.grid()\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.05))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "c_0 = 1.0\n",
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "n_steps = 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "t_exact = numpy.linspace(0.0, 1.6e3, 100)\n",
    "\n",
    "# Implement backwards Euler\n",
    "t_backwards = numpy.linspace(0.0, 1.6e3, n_steps)\n",
    "delta_t = t_backwards[1] - t_backwards[0]\n",
    "u_backwards = numpy.empty(t_backwards.shape)\n",
    "u_backwards[0] = c_0\n",
    "for n in range(0, t_backwards.shape[0] - 1):\n",
    "    u_backwards[n + 1] = u_backwards[n] / (1.0 - decay_constant * delta_t)\n",
    "\n",
    "fig = plt.figure(figsize=(16,6))\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "axes.plot(t_backwards, u_backwards, 'or', label=\"Backwards Euler\")\n",
    "axes.plot(t_exact, u_exact(t_exact), 'k--', label=\"True Solution\")\n",
    "axes.grid()\n",
    "axes.set_title(\"Backwards Euler\")\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_ylabel(\"$c(t)$\")\n",
    "axes.set_ylim((0.4,1.1))\n",
    "axes.legend()\n",
    "\n",
    "abs_err = numpy.abs(u_backwards - u_exact(t_backwards))\n",
    "rel_err = abs_err/u_exact(t_backwards)\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "axes.plot(t_backwards,abs_err,'ro',label='absolute error')\n",
    "axes.plot(t_backwards,rel_err,'bo',label='relative error')\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_ylabel(\"error\")\n",
    "axes.set_title('Error')\n",
    "axes.legend(loc='best')\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "It's also useful to be able to do this in the case of systems of ODEs.  Let $f(U) = A U$, then\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    U_{n+1} &= U_n + \\Delta t (A U_{n+1}) \\\\\n",
    "     \\left [ I - \\Delta t A \\right ]U_{n+1} &= U_n \\\\\n",
    "    U_{n+1} &= \\left [ I - \\Delta t A \\right]^{-1} U_n\n",
    "\\end{aligned}$$\n",
    "\n",
    "In general however we are often not able to do this with arbitrary $f$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Another simple implicit method is based on quadrature using the trapezoidal method.  The scheme is\n",
    "$$\n",
    "    \\frac{U_{n+1} - U_{n}}{\\Delta t} = \\frac{1}{2} (f(U_n) + f(U_{n+1}))\n",
    "$$\n",
    "\n",
    "In this case what is the update scheme for $f(u) = \\lambda u$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\\begin{aligned}\n",
    "    U_{n+1} &= U_{n} + \\frac{\\Delta t}{2} (f(U_n) + f(U_{n+1})) \\\\\n",
    "    U_{n+1} &= U_{n} + \\frac{\\Delta t}{2} (\\lambda U_n + \\lambda U_{n+1}) \\\\\n",
    "    U_{n+1} \\left[1 - \\frac{\\Delta t \\lambda}{2}  \\right] &= U_{n} \\left[1 + \\frac{\\Delta t \\lambda}{2} \\right] \\\\\n",
    "    U_{n+1} &= U_{n} \\frac{1 + \\frac{\\Delta t \\lambda}{2}}{1 - \\frac{\\Delta t \\lambda}{2}} \\\\\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "n_steps = 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "c_0 = 1.0\n",
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "t_exact = numpy.linspace(0.0, 1.6e3, 100)\n",
    "\n",
    "# Implement trapezoidal method\n",
    "t = numpy.linspace(0.0, 1.6e3, n_steps)\n",
    "delta_t = t[1] - t[0]\n",
    "u = numpy.empty(t.shape)\n",
    "u[0] = c_0\n",
    "integration_constant = (1.0 + decay_constant * delta_t / 2.0) / (1.0 - decay_constant * delta_t / 2.0)\n",
    "for n in range(t.shape[0] - 1):\n",
    "    u[n + 1] = u[n] * integration_constant\n",
    "\n",
    "fig = plt.figure(figsize=(16,6))\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "axes.plot(t, u, 'or', label=\"Trapezoidal\")\n",
    "axes.plot(t_exact, u_exact(t_exact), 'k--', label=\"True Solution\")\n",
    "axes.grid()\n",
    "\n",
    "axes.set_title(\"Trapezoidal\")\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_xlabel(\"$c(t)$\")\n",
    "axes.set_ylim((0.4,1.1))\n",
    "axes.legend()\n",
    "\n",
    "abs_err = numpy.abs(u  - u_exact(t))\n",
    "rel_err = abs_err/u_exact(t)\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "axes.plot(t,abs_err,'ro',label='absolute error')\n",
    "axes.plot(t,rel_err,'bo',label='relative error')\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_ylabel(\"error\")\n",
    "axes.set_title('Error')\n",
    "axes.legend(loc='best')\n",
    "axes.grid()\n",
    "\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Error Analysis of ODE Methods\n",
    "\n",
    "At this point it is also helpful to introduce more notation to distinguish between the true solution to the ODE $u(t_n)$ and the approximated value which we will denote $U_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Definition:** We define the *truncation error* of a scheme by replacing the $U_n$ with the true solution $u(t_n)$ in the finite difference formula and looking at the difference from the exact solution.\n",
    "\n",
    "For example we will use the difference form of forward Euler\n",
    "$$\n",
    "    \\frac{U_{n+1} - U_n}{\\Delta t} = f(t_n,U_n)\n",
    "$$\n",
    "and define the truncation error as\n",
    "$$\n",
    "    T(t, u; \\Delta t) = \\frac{u(t_{n+1}) - u(t_n)}{\\Delta t} - f(t_n, u(t_n)).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Definition:** A method is called *consistent* if \n",
    "$$\n",
    "    \\lim_{\\Delta t \\rightarrow 0} T(t, u; \\Delta t) = 0.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Definition:** We say that a method is *order* $p$ accurate if\n",
    "\n",
    "$$\n",
    "    \\lVert T(t, u; \\Delta t) \\rVert \\leq C \\Delta t^p\n",
    "$$\n",
    "\n",
    "uniformally on $t \\in [0, \\tau]$.  This can also be written as $T(t, u; \\Delta t) = \\mathcal{O}(\\Delta t^p)$.  Note that a method is consistent if $p > 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Error Analysis of Forward Euler\n",
    "\n",
    "We can analyze the error and convergence order of forward Euler by considering the Taylor series centered at $t_n$:\n",
    "\n",
    "$$\n",
    "    u(t) = u(t_n) + (t - t_n) u'(t_n) + \\frac{u''(t_n)}{2} (t - t_n)^2 + \\mathcal{O}((t-t_n)^3)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Evaluating this series at $t_{n+1}$ gives\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    u(t_{n+1}) &= u(t_n) + (t_{n+1} - t_n) u'(t_n) + \\frac{u''(t_n)}{2} (t_{n+1} - t_n)^2 + \\mathcal{O}((t_{n+1}-t_n)^3)\\\\\n",
    "    &=u_n + \\Delta t f(t_n, u_n) + \\frac{u''(t_n)}{2} \\Delta t^2 + \\mathcal{O}(\\Delta t^3)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "From the definition of truncation error we can use our Taylor series expression and find the truncation error.  Take the finite difference form of forward Euler\n",
    "\n",
    "$$\n",
    "    \\frac{U_{n+1} - U_n}{\\Delta t} = f(t_n, U_n)\n",
    "$$\n",
    "\n",
    "and replacing the derivative formulation with $u(t_n)$ to find\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    T(t, u; \\Delta t) &= \\frac{u(t_{n+1}) - u(t_n)}{\\Delta t} - f(t_n, u_n) \\\\\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Given the Taylor's series expansion for $u(t_{n+1})$\n",
    "\n",
    "$$\n",
    "u(t_{n+1}) =u(t_n) + \\Delta t f(t_n, u_n) + \\frac{u''(t_n)}{2} \\Delta t^2 + \\mathcal{O}(\\Delta t^3)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We substitute to find \n",
    "$$\n",
    "    T(t, u; \\Delta t)  = \\frac{u''(t_n)}{2} \\Delta t + \\mathcal{O}(\\Delta t^2).\n",
    "$$\n",
    "\n",
    "This implies that forward Euler is first order accurate and therefore consistent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Another equivalent definition of the truncation error uses the form\n",
    "$$\n",
    "    U_{n+1} = u(t_n) + \\Delta t f(t_n)\n",
    "$$\n",
    "and the definition\n",
    "$$\n",
    "    T(t, u; \\Delta t) = \\frac{1}{\\Delta t} \\left [ U_{n+1} - u(t_{n+1}) \\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "to find\n",
    "$$\\begin{aligned}\n",
    "    T(t, u; \\Delta t) &= \\frac{1}{\\Delta t} [U_{n+1} - u(t + \\Delta t)] \\\\\n",
    "    &= \\frac{1}{\\Delta t} \\left[ \\underbrace{u_n + \\Delta t f(t_n, u_n)}_{U_{n+1}} - \\underbrace{\\left( u_n + \\Delta t f(t_n, u_n) + \\frac{u''(t_n)}{2} \\Delta t^2 + \\mathcal{O}(\\Delta t^3) \\right )}_{u(t_{n+1})}\\right ] \\\\\n",
    "    &= \\frac{1}{\\Delta t} \\left[ - \\frac{u''(t_n)}{2} \\Delta t^2 - \\mathcal{O}(\\Delta t^3) \\right ] \\\\\n",
    "    &= - \\frac{u''(t_n)}{2} \\Delta t - \\mathcal{O}(\\Delta t^2)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Truncation Error vs Step Error\n",
    "\n",
    "Sometimes we will also consider the \"Step Error\"  which is the error that is introduced over one step\n",
    "\n",
    "$$\n",
    "    E_h = | U_{n+1} - u_{n+1} |\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This leads to an alternate definition of the truncation error as\n",
    "\n",
    "$$\n",
    "    T(t,u;\\Delta t) = \\frac{E_h}{\\Delta t} = \\frac{1}{\\Delta t} [U_{n+1} - u_{n+1}]\n",
    "$$\n",
    "\n",
    "so if the Truncation error is $O(\\Delta t^p)$ then the step error will be order $O(\\Delta t^{p+1})$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "So for Forward (or Backward's) Euler the step error \n",
    "\n",
    "$$\n",
    "    E_h = O(\\Delta t^2)\n",
    "$$\n",
    "\n",
    "The step error can be very useful in *adaptive stepping* schemes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Runge-Kutta Methods\n",
    "\n",
    "One way to derive higher-order ODE solvers is to use higher order quadrature schemes that sample the function at a number of  intermediate stages to provide a more accurate estimate of $K$.  These are not *multi-step* methods as they still only require information from the current time step but they raise the order of accuracy by adding *stages*.  These types of methods are called **Runge-Kutta** methods."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example:  Two-stage Runge-Kutta Methods\n",
    "\n",
    "The basic idea behind the simplest of the Runge-Kutta methods is to approximate $K$ using a mid-point scheme (which should be 2nd order accurate).  Unforrunately, we don't know the value of the mid-point.  However we can use an Euler step of size $\\Delta t/2$ to estimate the mid-point.  \n",
    "\n",
    "We can write the algorithm as \n",
    "\n",
    "$$\\begin{aligned}\n",
    "    K_1 &= \\Delta t f(U_n, t_n) \\\\\n",
    "    K_2 &= \\Delta t f(U_n + K_1/2, t_n + \\Delta t/2 )\\\\\n",
    "    U_{n+1} &= U_n + K_2 \\\\    \n",
    "\\end{aligned}$$\n",
    "\n",
    "Where we now evaluate the function in two stages $K_1$ and $K_2$.\n",
    "\n",
    "or for an autonomous ODE\n",
    "$$\n",
    "    U_{n+1} = U_n + \\Delta t f(U_n + \\frac{1}{2} \\Delta t f(U_n))\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "\n",
    "# RK2 step\n",
    "dt = 1e3\n",
    "U0 = 1.0\n",
    "K1 = dt * f(0., U0)\n",
    "Y1 = U0 + K1/2\n",
    "K2 = dt * f(dt/2., Y1)\n",
    "U1 = U0 + K2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, U0 * numpy.exp(decay_constant * t), label=\"True Solution\")\n",
    "axes.plot(0., U0, 'ro')\n",
    "axes.text(0., U0+.01, '$U_0$', fontsize=16)\n",
    "axes.plot((0.0, dt), (U0, U0 + K1), 'k--')\n",
    "axes.plot((0.0, dt/2.), (U0, U0 + K1/2.), 'k')\n",
    "\n",
    "axes.plot(dt/2., U0 + K1/2, 'ro')\n",
    "axes.plot((0.0, 0.0), (U0, Y1), 'k--')\n",
    "axes.text(10., 0.85, '$\\\\frac{K_1}{2}$', fontsize=18)\n",
    "axes.plot((0.0, dt/2), (Y1, Y1), 'k--')\n",
    "axes.text(250, Y1 - 0.05, '$\\\\frac{\\Delta t}{2}$', fontsize=18)\n",
    "\n",
    "axes.plot(dt, U1, 'go')\n",
    "axes.plot((0.0, 0.0), (U0, Y1), 'k--')\n",
    "axes.text(10., 0.85, '$\\\\frac{K_1}{2}$', fontsize=18)\n",
    "axes.plot((0.0, dt/2), (Y1, Y1), 'k--')\n",
    "axes.text(250, Y1 - 0.05, '$\\\\frac{\\Delta t}{2}$', fontsize=18)\n",
    "\n",
    "axes.plot(dt, U1, 'go')\n",
    "axes.plot((0., dt), (U0, U1), 'k')\n",
    "axes.text(dt+20, U1, '$U_1$', fontsize=18)\n",
    "#axes.plot((0.0, 0.0), (U0, U1), 'g--')\n",
    "#axes.plot((0.0, dt), (U1, U1), 'g--')\n",
    "\n",
    "\n",
    "\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.05))\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Error analysis RK2\n",
    "\n",
    "The truncation error can be computed similarly to how we did so before but we do need to figure out how to compute the derivative inside of the function.  Note that due to \n",
    "$$\n",
    "    f(u(t_n)) = u'(t_n)\n",
    "$$ \n",
    "that differentiating this with respect to $t$ leads to \n",
    "$$\n",
    "    f'(u(t_n)) u'(t_n) = u''(t_n)\n",
    "$$ \n",
    "leading to\n",
    "$$\\begin{aligned}\n",
    "    f\\left(u(t_n) + \\frac{1}{2} \\Delta t f(u(t_n)) \\right ) &= f\\left(u(t_n) +\\frac{1}{2} \\Delta t u'(t_n) \\right ) \\\\\n",
    "    &= f(u(t_n)) + \\frac{1}{2} \\Delta t u'(t_n) f'(u(t_n)) + \\frac{1}{8} \\Delta t^2 (u'(t_n))^2 f''(u(t_n)) + \\mathcal{O}(\\Delta t^3) \\\\\n",
    "    &=u'(t_n) + \\frac{1}{2} \\Delta t u''(t_n) + \\mathcal{O}(\\Delta t^2)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Using our alternative definition of the truncation error we have\n",
    "\n",
    "$$\n",
    "        T(t, u; \\Delta t) = \\frac{1}{\\Delta t} \\left[U_{n+1} - u_{n+1} \\right] \n",
    "$$\n",
    "or"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\begin{aligned}\n",
    "    T(t, u; \\Delta t) &= \\frac{1}{\\Delta t} \\left[u_n + \\Delta t f\\left(u_n + \\frac{1}{2} \\Delta t f(u_n)\\right) - \\left(u_n + \\Delta t f(t_n, u_n) + \\frac{u''(t_n)}{2} \\Delta t^2 + \\mathcal{O}(\\Delta t^3) \\right ) \\right] \\\\\n",
    "    &=\\frac{1}{\\Delta t} \\left[\\Delta t u'(t_n) + \\frac{1}{2} \\Delta t^2 u''(t_n) + \\mathcal{O}(\\Delta t^3) - \\Delta t u'(t_n) - \\frac{u''(t_n)}{2} \\Delta t^2 + \\mathcal{O}(\\Delta t^3) \\right] \\\\\n",
    "    &= \\mathcal{O}(\\Delta t^2)\n",
    "\\end{aligned}$$\n",
    "\n",
    "so this method is second order accurate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example:  Improved Euler's method\n",
    "\n",
    "The Improved Euler's method is another RK2 scheme but instead of approximating a mid-point quadrature rule, it approximates a trapezoidal rule.   \n",
    "\n",
    "We can write the algorithm as \n",
    "\n",
    "$$\\begin{aligned}\n",
    "    K_1 &= \\Delta t f(U_n, t_n) \\\\\n",
    "    K_2 &= \\Delta t f(U_n + K1, t_n + \\Delta t )\\\\\n",
    "    U_{n+1} &= U_n + \\frac{1}{2}\\left[K_1 +K_2\\right] \\\\    \n",
    "\\end{aligned}$$\n",
    "\n",
    "Where we now use function evaluations at both the initial value, and at the euler point but take the average of those slopes.\n",
    "\n",
    "Again, error analysis shows that this scheme also has a truncation error $T(t,u:\\Delta t) = \\mathcal{O}(\\Delta t^2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "\n",
    "# Improved Euler step\n",
    "dt = 1e3\n",
    "U0 = 1.0\n",
    "K1 = dt * f(0., U0)\n",
    "Y1 = U0 + K1\n",
    "K2 = dt * f(dt, Y1)\n",
    "U1 = U0 + 0.5*(K1 + K2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, U0 * numpy.exp(decay_constant * t), label=\"True Solution\")\n",
    "axes.plot(0., U0, 'ro')\n",
    "axes.text(0., U0+.01, '$U_0$', fontsize=16)\n",
    "axes.plot((0.0, dt), (U0, U0 + K1), 'k--')\n",
    "axes.plot((0.0, dt), (U0, U0 + K1), 'k')\n",
    "\n",
    "axes.plot(dt, Y1, 'ro')\n",
    "axes.text(dt+10, Y1, '$Y_1$', fontsize=18)\n",
    "axes.plot((0.0, 0.0), (U0, Y1), 'k--')\n",
    "axes.plot((0.0, dt), (Y1, Y1), 'k--')\n",
    "axes.text(350, Y1 - 0.05, '$\\\\frac{\\Delta t}{2}$', fontsize=18)\n",
    "\n",
    "axes.plot(dt, U1, 'go')\n",
    "axes.plot((0.0, 0.0), (U0, Y1), 'k--')\n",
    "axes.plot(0., Y1, 'ko')\n",
    "axes.text(10., Y1, '$K_1$', fontsize=18)\n",
    "\n",
    "axes.plot((0., 0.), (U0, U0+K2),'b--')\n",
    "axes.plot(0., U0+K2,'bo--')\n",
    "axes.text(10., U0+K2, '$K_2$', fontsize=18)\n",
    "\n",
    "\n",
    "axes.plot(0., U1,'gx', markersize=15)\n",
    "axes.text(10., U1, '$0.5*(K_1 +K_2)$', fontsize=18)\n",
    "\n",
    "\n",
    "axes.plot(dt, U1, 'go')\n",
    "axes.plot((0., dt), (U0, U1), 'k')\n",
    "axes.text(dt+20, U1, '$U_1$', fontsize=18)\n",
    "#axes.plot((0.0, 0.0), (U0, U1), 'g--')\n",
    "#axes.plot((0.0, dt), (U1, U1), 'g--')\n",
    "\n",
    "\n",
    "\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.05))\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example:  4-stage Runge-Kutta Method\n",
    "\n",
    "If RK2 is related to a Mid-point quadrature scheme,  then the classic 4-stage, 4th order Runge-Kutta scheme should be reminiscent of Simpson's Quadrature rule.  It requires 4 samples of $f(t,u)$ at the beginning of the step, two-samples in the middle and one at the end, then a linear combination of those samples\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    K_1 &= \\Delta t f(t_n, U_n) \\\\\n",
    "    K_2 &= \\Delta t f(t_n + \\Delta t/2, U_n + K_1/2) \\\\\n",
    "    K_3 &= \\Delta t f(t_n + \\Delta t/2, U_n + K_2/2) \\\\\n",
    "    K_4 &= \\Delta t f(t_n + \\Delta t, U_n + K_3) \\\\\n",
    "        & \\\\\n",
    "    U_{n+1} &= U_n + \\frac{1}{6} \\left [K_1 + 2(K_2 + K_3)  + K_4) \\right ] \n",
    "\\end{aligned}$$\n",
    "\n",
    "With truncation error $T = O(\\Delta t^4)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "\n",
    "# RK4 step\n",
    "dt = 1e3\n",
    "U0 = 1.0\n",
    "K1 = dt * f(0., U0)\n",
    "K2 = dt * f(dt/2., U0 + K1/2)\n",
    "K3 = dt * f(dt/2., U0 + K2/2)\n",
    "K4 = dt * f(dt, U0 + K3)\n",
    "\n",
    "U1 = U0 + 1./6. *( K1 + 2 * (K2 + K3) + K4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, U0 * numpy.exp(decay_constant * t), label=\"True Solution\")\n",
    "axes.plot(0., U0, 'ro')\n",
    "axes.text(0.-20, U0-.04, '$K_1$', color='red',fontsize=16)\n",
    "\n",
    "axes.text(0., U0+.01, '$U_0$', fontsize=16)\n",
    "axes.plot((0.0, dt/2.), (U0, U0 + K1/2.), 'k--')\n",
    "axes.plot(dt/2., U0 + K1/2, 'ro')\n",
    "axes.text(dt/2-20, U0 + K1/2-.04, '$K_2$', color='red',fontsize=16)\n",
    "\n",
    "\n",
    "axes.plot((0.0, dt/2.), (U0, U0 + K2/2.), 'k--')\n",
    "axes.plot(dt/2., U0 + K2/2, 'ro')\n",
    "axes.text(dt/2-20, U0 + K2/2+.02, '$K_3$', color='red',fontsize=16)\n",
    "\n",
    "axes.plot((0.0, dt), (U0, U0 + K3), 'k--')\n",
    "axes.plot(dt, U0 + K3, 'ro')\n",
    "axes.text(dt-20, U0 + K3-.04, '$K_4$', color='red',fontsize=16)\n",
    "\n",
    "axes.plot(dt, U1, 'go')\n",
    "#axes.plot((0., dt), (U0, U1), 'k')\n",
    "axes.text(dt+20, U1, '$U_1$', fontsize=18)\n",
    "\n",
    "\n",
    "\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.05))\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def RK2(f, t_span, u0, N):\n",
    "    \"\"\" implement constant step size 2 stage Runge-Kutta Method RK2\"\"\"\n",
    "    \n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    delta_t = t[1] - t[0]\n",
    "    u = numpy.empty(t.shape)\n",
    "    u[0] = u0 \n",
    "    for (n, t_n) in enumerate(t[:-1]):\n",
    "        K_1 = delta_t * f(t_n, u[n])\n",
    "        K_2 = delta_t * f(t_n + delta_t/2., u[n] + K_1/2.)\n",
    "        u[n+1] = u[n] + K_2\n",
    "    return t, u\n",
    "\n",
    "def improved_euler(f, t_span, u0, N):\n",
    "    \"\"\" implement constant step size 2 stage Improved Euler Method trapezoidal rule\"\"\"\n",
    "    \n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    delta_t = t[1] - t[0]\n",
    "    u = numpy.empty(t.shape)\n",
    "    u[0] = u0\n",
    "    for (n, t_n) in enumerate(t[:-1]):\n",
    "        K_1 = delta_t * f(t_n, u[n])\n",
    "        K_2 = delta_t * f(t_n + delta_t, u[n] + K_1)\n",
    "        u[n+1] = u[n] + 0.5 * (K_1 + K_2)\n",
    "    return t, u "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def RK4(f, t_span, u0, N):\n",
    "    \"\"\" implement constant step size 4 stage Runge-Kutta Method RK4\"\"\"\n",
    "    \n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    delta_t = t[1] - t[0]\n",
    "    u = numpy.empty(t.shape)\n",
    "    u[0] = u0\n",
    "    for (n, t_n) in enumerate(t[:-1]):\n",
    "        K_1 = delta_t * f(t_n, u[n])\n",
    "        K_2 = delta_t * f(t_n + delta_t/2., u[n] + K_1/2.)\n",
    "        K_3 = delta_t * f(t_n + delta_t/2., u[n] + K_2/2.)\n",
    "        K_4 = delta_t * f(t_n + delta_t, u[n] + K_3)\n",
    "        u[n+1] = u[n] + 1./6. * (K_1 + 2.*( K_2 + K_3) + K_4)\n",
    "    return t, u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Implement and compare the two-stage and 4-stage Runge-Kutta methods \n",
    "f = lambda t, u: -u\n",
    "N = 20\n",
    "t_span = [ 0., 5.0 ]\n",
    "u0 = 1.\n",
    "\n",
    "u_exact = lambda t: u0*numpy.exp(-t)\n",
    "\n",
    "t_exact = numpy.linspace(t_span[0], t_span[1], 100)\n",
    "t_euler, u_euler = euler(f, t_span, u0, N)\n",
    "t_ieuler, u_ieuler = improved_euler(f, t_span, u0, N)\n",
    "t_RK2, u_RK2 = RK2(f, t_span, u0, N)\n",
    "t_RK4, u_RK4 = RK4(f, t_span, u0, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(16,6))\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "axes.plot(t_exact,u_exact(t_exact),'k',label='exact')\n",
    "axes.plot(t_euler, u_euler, 'ro', label='euler')\n",
    "axes.plot(t_ieuler, u_ieuler, 'co', label='improved euler')\n",
    "axes.plot(t_RK2, u_RK2, 'go', label='RK2')\n",
    "axes.plot(t_RK4, u_RK4, 'bo', label='RK4')\n",
    "\n",
    "axes.grid()\n",
    "axes.set_xlabel('t', fontsize=16)\n",
    "axes.set_ylabel('u', fontsize=16)\n",
    "axes.legend(loc='best')\n",
    "\n",
    "err = lambda u, t: numpy.abs(u - u_exact(t))/u_exact(t)\n",
    "\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "\n",
    "axes.semilogy(t_euler,err(u_euler,t_euler),'ro',label='euler')\n",
    "axes.semilogy(t_ieuler,err(u_ieuler,t_ieuler),'co',label='improved euler')\n",
    "axes.semilogy(t_RK2,err(u_RK2,t_RK2),'go',label='RK2')\n",
    "axes.semilogy(t_RK4,err(u_RK4,t_RK4),'bo',label='RK4')\n",
    "\n",
    "axes.set_xlabel(\"t (years)\")\n",
    "axes.set_ylabel(\"Rel. error\")\n",
    "axes.set_title('Error')\n",
    "axes.legend(loc='best')\n",
    "axes.grid()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Convergence of Single Step Multi-Stage schemes\n",
    "\n",
    "All of the above schemes are consistent and have truncation errors $T\\propto\\Delta t^p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "N = numpy.array([ 2**n for n in range(4,10)])\n",
    "err_euler = numpy.zeros(len(N))\n",
    "err_ieuler = numpy.zeros(len(N))\n",
    "err_RK2 = numpy.zeros(len(N))\n",
    "err_RK4 = numpy.zeros(len(N))\n",
    "\n",
    "t_span = [ 0., 4.]\n",
    "dt = t_span[1]/N\n",
    "\n",
    "u0 = 1. \n",
    "u_exact = u0*numpy.exp(-t_span[1])\n",
    "\n",
    "for i, n in enumerate(N):\n",
    "    t, u_euler = euler(f, t_span, u0, n)\n",
    "    err_euler[i] = numpy.abs(u_euler[-1] - u_exact)\n",
    "    t, u_ieuler = improved_euler(f, t_span, u0, n)\n",
    "    err_ieuler[i] = numpy.abs(u_ieuler[-1] - u_exact)\n",
    "    t, u_RK2 = RK2(f, t_span, u0, n)\n",
    "    err_RK2[i] = numpy.abs(u_RK2[-1] - u_exact)\n",
    "    t, u_RK4 = RK4(f, t_span, u0, n)\n",
    "    err_RK4[i] = numpy.abs(u_RK4[-1] - u_exact)\n",
    "    \n",
    "err_fit = lambda dt, p: numpy.exp(p[1])*dt**p[0]\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "# Euler\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_euler[2:]),1)\n",
    "line = axes.loglog(dt, err_euler, 'o', label='euler, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "# Improved Euler\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_ieuler[2:]),1)\n",
    "line = axes.loglog(dt, err_ieuler, 'o', label='improved euler, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "# RK2\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_RK2[2:]),1)\n",
    "line = axes.loglog(dt, err_RK2, 'o', label='rk2, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "#RK4\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_RK4[2:]),1)\n",
    "line = axes.loglog(dt, err_RK4, 'o', label='rk4, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "\n",
    "axes.grid()\n",
    "axes.set_xlabel('$\\Delta t$', fontsize=16)\n",
    "axes.set_ylabel('$Error$', fontsize=16)\n",
    "axes.set_title('Convergence: Single Step Schemes', fontsize=18)\n",
    "axes.legend(loc='best', fontsize=14)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Summary: single-Step Multi-Stage Schemes\n",
    "\n",
    "The integral form of an ODE initial value problem can be written\n",
    "\n",
    "$$ \n",
    "    u(t + \\Delta t) = u(t) + \\int^{t + \\Delta t}_t f(\\tau, u(\\tau)) d\\tau\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Which says that our solution $u$, if it exists  at some time $\\Delta t$ in the future,  is $u(t)$ plus a *number* \n",
    "\n",
    "$$\n",
    "    K = \\int^{t + \\Delta t}_t f(\\tau, u(\\tau) )d\\tau\n",
    "$$\n",
    "\n",
    "which is a definite *line integral* (along an unknown solution). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Single Step, Multi-stage schemes\n",
    "\n",
    "are most easily understood as extensions of the  Newton-Cotes quadrature schemes for approximating $K$ (plus an error term that will scale as $\\Delta t^p$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "**Explicit Schemes**\n",
    "<table width=\"80%\">\n",
    "    <tr align=\"center\"><th>Name</th> <th align=\"center\">Stages</th> <th align=\"center\">\"Quadrature\"</th><th align=\"center\">$$T$$</th></tr>\n",
    "     <tr align=\"center\"><td>Euler</td> <td align=\"center\">1</td> <td align=\"center\">Left-Rectangle</td><td align=\"center\">$$O(\\Delta t)$$</td></tr>\n",
    "    <tr align=\"center\"><td>Improved Euler</td> <td align=\"center\">2</td> <td align=\"center\">Trapezoidal</td><td align=\"center\">$$O(\\Delta t^2)$$</td></tr>\n",
    "    <tr align=\"center\"><td>RK2</td> <td align=\"center\">2</td> <td align=\"center\">Mid-Point</td><td align=\"center\">$$O(\\Delta t^2)$$</td></tr>\n",
    "    <tr align=\"center\"><td>RK4</td> <td align=\"center\">4</td> <td align=\"center\">Simpson</td><td align=\"center\">$$O(\\Delta t^4)$$</td></tr>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Implicit Schemes**\n",
    "<table width=\"80%\">\n",
    "    <tr align=\"center\"><th>Name</th> <th align=\"center\">Stages</th> <th align=\"center\">\"Quadrature\"</th><th align=\"center\">$$T$$</th></tr>\n",
    "     <tr align=\"center\"><td>Backwards-Euler</td> <td align=\"center\">1</td> <td align=\"center\">Right-Rectangle</td><td align=\"center\">$$O(\\Delta t)$$</td></tr>\n",
    "    <tr align=\"center\"><td>Trapezoidal</td> <td align=\"center\">2</td> <td align=\"center\">Trapezoidal</td><td align=\"center\">$$O(\\Delta t^2)$$</td></tr>\n",
    "</table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Convergence of Single Step Multi-Stage schemes\n",
    "\n",
    "All of the above schemes are consistent and have truncation errors $T\\propto\\Delta t^p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "N = numpy.array([ 2**n for n in range(4,10)])\n",
    "err_euler = numpy.zeros(len(N))\n",
    "err_ieuler = numpy.zeros(len(N))\n",
    "err_RK2 = numpy.zeros(len(N))\n",
    "err_RK4 = numpy.zeros(len(N))\n",
    "\n",
    "t_span = [ 0., 4.]\n",
    "dt = t_span[1]/N\n",
    "\n",
    "u0 = 1. \n",
    "u_exact = u0*numpy.exp(-t_span[1])\n",
    "\n",
    "for i, n in enumerate(N):\n",
    "    t, u_euler = euler(f, t_span, u0, n)\n",
    "    err_euler[i] = numpy.abs(u_euler[-1] - u_exact)\n",
    "    t, u_ieuler = improved_euler(f, t_span, u0, n)\n",
    "    err_ieuler[i] = numpy.abs(u_ieuler[-1] - u_exact)\n",
    "    t, u_RK2 = RK2(f, t_span, u0, n)\n",
    "    err_RK2[i] = numpy.abs(u_RK2[-1] - u_exact)\n",
    "    t, u_RK4 = RK4(f, t_span, u0, n)\n",
    "    err_RK4[i] = numpy.abs(u_RK4[-1] - u_exact)\n",
    "    \n",
    "err_fit = lambda dt, p: numpy.exp(p[1])*dt**p[0]\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "# Euler\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_euler[2:]),1)\n",
    "line = axes.loglog(dt, err_euler, 'o', label='euler, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "# Improved Euler\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_ieuler[2:]),1)\n",
    "line = axes.loglog(dt, err_ieuler, 'o', label='improved euler, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "# RK2\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_RK2[2:]),1)\n",
    "line = axes.loglog(dt, err_RK2, 'o', label='rk2, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "#RK4\n",
    "p = numpy.polyfit(numpy.log(dt[2:]), numpy.log(err_RK4[2:]),1)\n",
    "line = axes.loglog(dt, err_RK4, 'o', label='rk4, p={:3.2f}'.format(p[0]))\n",
    "axes.loglog(dt, err_fit(dt,p),'--', color=line[0].get_color())\n",
    "\n",
    "\n",
    "axes.grid()\n",
    "axes.set_xlabel('$\\Delta t$', fontsize=16)\n",
    "axes.set_ylabel('$Error$', fontsize=16)\n",
    "axes.set_title('Convergence: Single Step Schemes', fontsize=18)\n",
    "axes.legend(loc='best', fontsize=14)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## General Explicit Runge-Kutta Schemes and ``Butcher Tableaus''\n",
    "\n",
    "\n",
    "The most general form of an explicit RK scheme with $s$ stages is\n",
    "\n",
    "$$\n",
    "    u_{n+1} = u_n +  \\sum_{i=1}^s b_i K_{i}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "K_{i} = \\Delta t f\\left(t_n + c_i \\Delta t, u_n + \\sum_{j=1}^{i-1} a_{ij} K_{j}\\right)\n",
    "$$\n",
    "\n",
    "$K_{i}$ are the function evaluations at stage $i$ which are determined by coefficients $a_{ij}$, $b_i$ and $c_i$.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "All of the critical coefficients for any explicit RK scheme can be arranged in a ``Butcher tableau'' \n",
    "\n",
    "$$\\begin{array} {c|ccccc} 0 & 0\\\\ c_2 & a_{21}\\\\ c_3 & a_{31} & a_{32} \\\\ \\vdots & \\vdots & & \\ddots \\\\ c_s& a_{s1}& a_{s2} & \\cdots & a_{s,s-1}\\\\ \\hline & b_1 & b_2 & \\cdots & b_{s-1} & b_s \\end{array}\n",
    "$$\n",
    "\n",
    "where $c_i$ gives the fraction of the time step for each stage.  $b_i$ gives the weighting for each function evaluation $K_i$ in the general quadrature and the $a_{ij}$ determine how to use previous stages to advance the value of $u$ at stage $i$.\n",
    "\n",
    "$$\n",
    "K_{i} = \\Delta t f\\left(t_n + c_i \\Delta t, u_n + \\sum_{j=1}^{i-1} a_{ij} K_{j}\\right)\n",
    "$$\n",
    "\n",
    "for more information see [here](https://www.johndcook.com/blog/2020/02/13/runge-kutta-methods/#:~:text=For%20explicit%20Runge%2DKutta%20methods,Butcher%20who%20classified%20RK%20methods)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "All of the schemes we have discussed can be written as Butcher Tableaus (although they can be a bit hard to parse).  But for example:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Euler's method\n",
    "\n",
    "$$\n",
    "\\begin{array} {c|c} 0 & 0\\\\ \\hline & 1\\end{array}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Backwards Euler's method\n",
    "\n",
    "$$\n",
    "\\begin{array} {c|c} 1 & 1\\\\ \\hline & 1\\end{array}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Classical RK4\n",
    "\n",
    "$$\n",
    "\\begin{array} {c|cccc} 0\\\\ 1/2 & 1/2\\\\ 1/2 &0 &1/2 \\\\ 1& 0& 0& 1\\\\ \\hline & 1/6 & 1/3 & 1/3 &1/6 \\end{array}\n",
    "$$\n",
    "\n",
    "Wikipedia provides a convenient [list](https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods)  of a large range of explicit and implicit RK schemes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Adaptive Time Stepping"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Why should we care about all of these schemes and their errors?\n",
    "\n",
    "* Even though we know the formal error.  It is with respect to a true solution we don't know.\n",
    "* In itself, the error estimates don't tell us how to choose a time step $\\Delta t$ to keep the solution within a given tolerance\n",
    "* However,  in combination, we can use multiple methods to control the error and provide **Adaptive** time stepping"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Example:  Compare 1 step of Euler to one step of RK2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "\n",
    "# RK2 step\n",
    "dt = 1e3\n",
    "U0 = 1.0\n",
    "K1 = dt * f(0., U0)\n",
    "Y1 = U0 + K1/2\n",
    "K2 = dt * f(dt/2., Y1)\n",
    "U1 = U0 + K2\n",
    "\n",
    "t = numpy.linspace(U0, 1600.)\n",
    "u_true = U0 * numpy.exp(decay_constant * t)\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, u_true, label=\"True Solution\")\n",
    "axes.plot(0., U0, 'ro')\n",
    "axes.text(0., U0+.01, '$U_0$', fontsize=16)\n",
    "axes.plot((0.0, dt), (U0, U0 + K1), 'k')\n",
    "#axes.plot((0.0, dt/2.), (U0, U0 + K1/2.), 'k')\n",
    "\n",
    "#Euler step\n",
    "axes.plot(dt, U0 + K1, 'ro')\n",
    "#axes.plot((0.0, 0.0), (U0, Y1), 'k--')\n",
    "axes.text(dt + 10., U0 + K1, '$U_{euler}$', fontsize=18)\n",
    "\n",
    "axes.plot(dt, U1, 'go')\n",
    "#axes.plot((0.0, 0.0), (U0, Y1), 'k--')\n",
    "#axes.text(10., 0.85, '$\\\\frac{K_1}{2}$', fontsize=18)\n",
    "#axes.plot((0.0, dt/2), (Y1, Y1), 'k--')\n",
    "#axes.text(250, Y1 - 0.05, '$\\\\frac{\\Delta t}{2}$', fontsize=18)\n",
    "\n",
    "# RK2 Step\n",
    "axes.plot(dt, U1, 'go')\n",
    "axes.plot((0., dt), (U0, U1), 'k')\n",
    "axes.text(dt+20, U1, '$U_{RK2}$', fontsize=18)\n",
    "#axes.plot((0.0, 0.0), (U0, U1), 'g--')\n",
    "#axes.plot((0.0, dt), (U1, U1), 'g--')\n",
    "\n",
    "# difference\n",
    "axes.plot((dt, dt), (U1, U0+K1),'k--')\n",
    "axes.text(dt+40, 0.5*(U1 + U0+K1), '$\\Delta\\propto\\Delta t^2$', fontsize=18)\n",
    "\n",
    "\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.05))\n",
    "axes.legend(loc='best')\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Relative Truncation Error \n",
    "\n",
    "If we consider the *Step Error* for each of our schemes, we know that\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    U^{euler}_{n+1}  &= u_{n+1} + O(\\Delta t^2)\\\\\n",
    "    U^{RK2}_{n+1} &= u_{n+1}  + O(\\Delta t^3)\\\\\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Therefore we can compute the *relative truncation error* as\n",
    "\n",
    "$$\n",
    "    \\Delta = | U^{euler}_{n+1} - U^{RK2}_{n+1} | = O(\\Delta t^{?})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "* $\\Delta$ is Computable!\n",
    "* has a known dependence on step-size $\\Delta t$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Adaptive Time Stepping\n",
    "\n",
    "Given the relative truncation error and its scaling with $\\Delta t$, we can now use this to choose a single good time step."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Example:\n",
    "\n",
    "Suppose we wanted our relative truncation error to be small relative to the solution or zero, we could set \n",
    "\n",
    "$$\n",
    "    \\Delta_{target} = \\mathtt{rtol}\\,U^{RK2}_{n+1} + \\mathtt{atol}\n",
    "$$\n",
    "\n",
    "where $\\mathtt{rtol}$ and $\\mathtt{atol}$ are relative and absolute tolerances (and we assume that $U^{RK2}_{n+1}$ is a reasonably good estimate of the true solution)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Moreover,  we know how the relative truncation error should scale with time step, i.e.\n",
    "\n",
    "$$\n",
    "    \\Delta_{target} \\propto \\Delta t_{target}^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "\n",
    "But our measured relative truncation error, $\\Delta$ depends on  the step size we just took i.e\n",
    "\n",
    "$$\n",
    "    \\Delta_{measured} \\propto \\Delta t_n^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Adaptive Time Stepping\n",
    "\n",
    "If we take the ratio of both relationships we get\n",
    "\n",
    "$$\n",
    "    \\frac{\\Delta_{target}}{\\Delta_{measured}} = \\left[\\frac{\\Delta t_{target}}{\\Delta t_{n}}\\right]^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "or rearranging, we can solve for the target step size\n",
    "\n",
    "$$\n",
    "    \\Delta t_{target} = \\Delta t_{n}\\left[\\frac{\\Delta_{target}}{\\Delta_{measured}}\\right]^{\\frac{1}{2}}\n",
    "$$\n",
    "\n",
    "which tells us how to grow or shrink our time step to maintain accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In general,  if we have two methods with different step errors such that \n",
    "\n",
    "$$\n",
    "    \\Delta \\propto \\Delta t^p\n",
    "$$\n",
    "\n",
    "then our adaptive stepper will look like\n",
    "\n",
    "$$\n",
    "    \\Delta t_{target} = \\Delta t_{n}\\left[\\frac{\\Delta_{target}}{\\Delta_{measured}}\\right]^{1/p}\n",
    "$$\n",
    "\n",
    "This leads to all sorts of adaptive schemes most are included in standard libraries.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Embedded Runge-Kutta Schemes\n",
    "\n",
    "There are in fact a whole family of **Embedded RK** schemes which are $N$ stage schemes but can combine the $N$ function evaluations in two different ways to produce methods with different error estimates.  \n",
    "\n",
    "A popular one is **RK45** (available in `SciPy`) which is based on the [Dormand-Prince 5(4)](https://doi.org/10.1016/0771-050X(80)90013-3) pair which uses 6 function evaluations per step to produce a 4th order and 5th order scheme.  The 4th order scheme controls the time step, and the 5th order scheme actually is the solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "#### Dormand-Prince 5(4) Butcher Tableau\n",
    "\n",
    "$$\n",
    "\\begin{array} {c|ccccccc} \n",
    "0\\\\ \n",
    "1/5 & 1/5\\\\ \n",
    "3/10 &3/40 & 9/40 \\\\ \n",
    "4/5 & 44/45& & −56/15\t&32/9\\\\ \n",
    "8/9\t& 19372/6561\t& −25360/2187\t& 64448/6561 &\t−212/729\\\\\n",
    "1 & 9017/3168 &\t−355/33\t& 46732/5247 & 49/176& −5103/18656 \\\\\n",
    "1 & 35/384 & 0\t& 500/1113\t& 125/192\t&−2187/6784\t& 11/84\t\\\\ \\hline\n",
    "& 35/384 &\t0\t& 500/1113 &\t125/192 &\t−2187/6784& \t11/84\t&0\\\\\n",
    "& 5179/57600 &\t0\t& 7571/16695\t& 393/640\t& −92097/339200\t& 187/2100\t& 1/40\\\\\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "\n",
    "The first row of $b$ coefficients gives the fifth-order accurate solution and the second row gives the fourth-order accurate solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import solve_ivp\n",
    "\n",
    "def f_vanderpol(t, u, mu=5):\n",
    "    return numpy.array([u[1], mu * (1.0 - u[0]**2) * u[1] - u[0]])\n",
    "\n",
    "t_span = (0., 50.)\n",
    "u0 = [ 1., 0. ]\n",
    "f = lambda t, u : f_vanderpol(t, u, mu=20) \n",
    "sol = solve_ivp(f, t_span, u0, method='Radau',rtol=1.e-3,atol=1.e-8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(16,6))\n",
    "axes = fig.add_subplot(1, 2, 1)\n",
    "\n",
    "axes.plot(sol.t, sol.y[0],'o-')\n",
    "axes.set_title(\"Solution to Van der Pol Oscillator\", fontsize=18)\n",
    "axes.set_xlabel(\"t\", fontsize=16)\n",
    "axes.set_ylabel(\"y(t)\", fontsize=16)\n",
    "axes.grid()\n",
    "\n",
    "axes = fig.add_subplot(1, 2, 2)\n",
    "delta_t = sol.t[1:] - sol.t[:-1]\n",
    "axes.plot(sol.t[:-1], delta_t)\n",
    "axes.grid()\n",
    "axes.set_xlabel('$t$', fontsize=16)\n",
    "axes.set_ylabel('$\\Delta t$', fontsize=16)\n",
    "axes.set_title('Timesteps, N = {}'.format(len(sol.t)), fontsize=18)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Taylor Series Methods\n",
    "\n",
    "A **Taylor series method** can be derived by direct substitution of the right-hand-side function $f(t, u)$ and its appropriate derivatives into the Taylor series expansion for $u(t_{n+1})$.  For a $p$th order method we would look at the Taylor series up to that order and replace all the derivatives of $u$ with derivatives of $f$ instead.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "For the general case we have\n",
    "$$\\begin{align*}\n",
    "    u(t_{n+1}) = u(t_n) + \\Delta t u'(t_n) + \\frac{\\Delta t^2}{2} u''(t_n) + \\frac{\\Delta t^3}{6} u'''(t_n) + \\cdots + \\frac{\\Delta t^p}{p!} u^{(p)}(t_n)\n",
    "\\end{align*}$$\n",
    "which contains derivatives of $u$ up to $p$th order.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We then replace these derivatives with the appropriate derivative of $f$ which will always be one less than the derivative of $u$ (due to the original ODE)\n",
    "\n",
    "$$\n",
    "    u^{(p)}(t_n) = f^{(p-1)}(t_n, u(t_n))\n",
    "$$\n",
    "\n",
    "leading to the method\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    u(t_{n+1}) &= u(t_n) + \\Delta t f(t_n, u(t_n)) + \\frac{\\Delta t^2}{2} f'(t_n, u(t_n)) \\\\\n",
    "    &+ \\frac{\\Delta t^3}{6} f''(t_n, u(t_n)) + \\cdots + \\frac{\\Delta t^p}{p!} f^{(p-1)}(t_n, u(t_n)).\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### 2nd Order Taylor Series Method\n",
    "\n",
    "We want terms up to second order so we need to take the derivative of $u' = f(t, u)$ once to find $u'' = f'(t, u)$ and therefore\n",
    "$$\\begin{align*}\n",
    "    u(t_{n+1}) &= u(t_n) + \\Delta t u'(t_n) + \\frac{\\Delta t^2}{2} u''(t_n) \\\\\n",
    "    &=u(t_n) + \\Delta t f(t_n, u(t_n)) + \\frac{\\Delta t^2}{2} f'(t_n, u(t_n)) ~~~ \\text{or} \\\\\n",
    "    U_{n+1} &= U_n + \\Delta t f(t_n, U_n) + \\frac{\\Delta t^2}{2} f'(t_n, U_n).\n",
    "\\end{align*}$$\n",
    "\n",
    "With Step error $O(\\Delta t^3)$ and truncation error $T$, $O(\\Delta t^2)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example\n",
    "\n",
    "Let's use our simplest problem $u'(t) = \\lambda u$ with $f=\\lambda u$.  Therefore\n",
    "\n",
    "$$\\begin{align*}\n",
    "    f(t,u) &= \\lambda u\\\\\n",
    "    f'(t,u) &= \\lambda u' = \\lambda f = \\lambda^2 u\\\\\n",
    "    f''(t,u) &= \\lambda^2 u' = \\lambda^2 f = \\lambda^3 u\n",
    "\\end{align*}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "so a third order scheme would look like\n",
    "$$\n",
    "\\begin{align}\n",
    "    U(t_{n+1}) &= U(t_n)\\left[ 1 + \\lambda\\Delta t  + \\frac{(\\lambda\\Delta t)^2}{2} + \\frac{(\\lambda\\Delta t)^3}{6}\\right]+ O(\\Delta t^4)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def Taylor_3_flambda_u(lamda, t_span, u0, N):\n",
    "    \"\"\" implement constant step size  3rd order Taylor Series method for f(t,u) = \\lambda u\"\"\"\n",
    "    \n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    lambda_dt = lamda*(t[1] - t[0])\n",
    "    u = numpy.empty(t.shape)\n",
    "    u[0] = u0 \n",
    "    for (n, t_n) in enumerate(t[:-1]):\n",
    "        u[n+1] = u[n] * ( 1. + lambda_dt  + (lambda_dt**2)/2.  + (lambda_dt**3)/6.)\n",
    "    return t, u\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "lam = -1.\n",
    "t_span = [0., 5.]\n",
    "u0 = 1.\n",
    "\n",
    "f = lambda t,u : -u\n",
    "t_exact = numpy.linspace(t_span[0], t_span[1], 100)\n",
    "u_exact = u0*numpy.exp(-t_exact)\n",
    "\n",
    "N = 20\n",
    "t_taylor, u_taylor = Taylor_3_flambda_u(lam, t_span, u0, N)\n",
    "t_euler, u_euler = euler(f, t_span, u0, N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t_exact,u_exact,'k',label='exact')\n",
    "axes.plot(t_euler, u_euler, 'ro', label='euler')\n",
    "axes.plot(t_taylor, u_taylor, 'bo', label='Taylor3')\n",
    "\n",
    "axes.grid()\n",
    "axes.set_xlabel('t', fontsize=16)\n",
    "axes.set_ylabel('u', fontsize=16)\n",
    "axes.legend(loc='best')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Some Drawbacks\n",
    "\n",
    "**Taylor Series methods**\n",
    " - require differentiating the given equation which can be cumbersome and difficult to implement\n",
    " - require a new routine for every $f$\n",
    " \n",
    "**General one-step/multi-stage methods**\n",
    "  - higher order methods often require a large number of evaluations of $f$ per time step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Overview -- so far\n",
    "\n",
    "So far we have discussed 3 basic techniques for integration of ODE IVP's\n",
    "\n",
    "* **Single-Step Multi-Stage** schemes (explicit and implicit)\n",
    "* **Taylor's Series** Methods\n",
    "* **Linear Multi-step** schemes (just started this)\n",
    "\n",
    "as well as \n",
    "* **truncation error** of each method (and it's relation to step-error)\n",
    "* **adaptive stepping** for Single-Step Schemes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In general Single-Step Multi-Stage methods (e.g. Embedded RK schemes) plus adaptive time stepping make for a very robust family of solvers. However there are some other classical schemes worth mentioning that have some advantages"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Linear Multi-Step Methods\n",
    "\n",
    "**Multi-step methods**  are ODE methods that \n",
    " - require only *one* new function evaluation per time step  to work.  \n",
    " - reuse values and function evaluations at some number of previous time steps\n",
    " \n",
    "**Disadvantages over single step methods**\n",
    "\n",
    " - Methods are not self-starting, i.e. they require other methods to find the initial values\n",
    " - Difficult to adapt. The time step  Δ𝑡  in one-step methods can be changed at any time while multi-step methods this is much more complex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Simplest example:  The leap-frog method\n",
    "\n",
    "The leap-frog method is similar to Euler's method in that it uses the information from two-previous time steps to advance the problem.  We can write the problem as a centered first-derivative centered   at time $t_{n+1}, U_{n+1}$ i.e. \n",
    "\n",
    "$$\\frac{U_{n+2} - U_{n}}{2\\Delta t} = f(t_{n+1}, U_{n+1})$$\n",
    "\n",
    "or\n",
    "\n",
    "$$\n",
    "    U_{n+2} = U_{n} + 2\\Delta t\\, f(t_{n+1}, U_{n+1})\n",
    "$$ \n",
    "\n",
    "this method is known as the leap-frog method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = numpy.linspace(0.0, 1.6e3, 100)\n",
    "c_0 = 1.0\n",
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t, c_0 * numpy.exp(decay_constant * t), label=\"True Solution\")\n",
    "\n",
    "# Plot Leap-Frog step\n",
    "dt = 1e3\n",
    "u1 = c_0 * numpy.exp(decay_constant * dt / 2.0)\n",
    "u_np = c_0 + dt * (decay_constant * u1)\n",
    "axes.plot((0.0, dt), (c_0, u_np), 'k')\n",
    "axes.plot((dt, dt), (u_np, c_0 * numpy.exp(decay_constant * dt)), 'k--')\n",
    "axes.plot((0.0, 0.0), (c_0, u_np), 'k--')\n",
    "axes.plot((0.0, dt), (u_np, u_np), 'k--')\n",
    "axes.text(400, u_np - 0.05, '$\\Delta t$', fontsize=16)\n",
    "axes.plot([0., dt/2, dt], [ c_0, u1, u_np],'ro')\n",
    "axes.set_title(\"Radioactive Decay with $t_{1/2} = 1600$ years\")\n",
    "axes.set_xlabel('t (years)')\n",
    "axes.set_ylabel('$c$')\n",
    "axes.grid()\n",
    "axes.set_xlim(-1e2, 1.6e3)\n",
    "axes.set_ylim((0.5,1.0))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def leap_frog(f, t_span, u0, N, start=RK2):\n",
    "    \"\"\" calculate fixed step with leap-frog iterator with a single step starter\n",
    "    \"\"\"\n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    delta_t = t[1] - t[0]\n",
    "    u = numpy.zeros(t.shape)\n",
    "    u[0] = u0\n",
    "    # use a single-step multi-stage method to start\n",
    "    \n",
    "    t_start, u_start = start(f, (t[0],t[1]), u0, 2)\n",
    "    u[1] = u_start[-1]\n",
    "    for (n, t_np) in enumerate(t[1:-1]):\n",
    "        u[n+2] = u[n] +  2 *delta_t * f(t_np, u[n+1]) \n",
    "    return t, u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "u0 = 1.0\n",
    "t_span = (0., 1600.) \n",
    "N = 7\n",
    "\n",
    "# Stable example\n",
    "decay_constant = -numpy.log(2.0) / 1600.0\n",
    "f = lambda t, u: decay_constant * u\n",
    "\n",
    "t_exact = numpy.linspace(t_span[0], t_span[1], N)\n",
    "u_exact = u0 * numpy.exp( decay_constant * t_exact)\n",
    "\n",
    "t_leapfrog, u_leapfrog = leap_frog(f, t_span, u0, N, start=RK4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9e7j+eo5Pm0ZECleDEREJdR999BEfffTRGZd/+eWXZ1xWqVKlkydUndCuXbvTxhUsWPC0M6zPJjY29uT7wMnVq1cvxWVZs2Zl4MCBZz1Lu3nz5jRv3vzk9KBBg8idOzf58+dPdW3BIEPsGZ9QrGlTbNYsVh87RoUqVfhNN5QQEQlpQ4YMYf78+WzatIkxY8bQt29fOnToEHK3vwytagOhfHkyjRnDfqDBPffw67BhXlckIiIXaP369dx2222UK1eO5557jq5du/LGG294XdZ5yzCHqZMq2bAhs2bN4oZ69bixSxe+OXCAeo895nVZIiIh4UyHm70wYMAABgwY4HUZFy3j7Rn7xdSpw6z584nJmpUm//sfv771ltcliYhIBpVhwxjg8sqVmbl0KR3z5aPS00/D+PFelyQiIhlQhg5jgHxlyjBk3TqiatRgT+vWfPPEE16XJCIiGUyGD2MA8uSBadPoW7gwLd58kxHt23tdkYiIZCAK4xNy5qTvkiU0yZ+fB0aNYlDLll5XJCIiGYTCOInsefMyYeNGbi9UiB4TJvByo0ZelyQiIhmAwjiZLDlzMnbDBu654gqGfv89/z72GATRafwiIhlJvXr16Nat20WvZ8aMGZgZf/31VwCqCjyFcQoyZcvGx2vXMq9NGy4dOJDjjz6KO37c67JERM7KzM766NChQ7rW89VXX1G7dm3y5MlDzpw5KVu2LJ06dUrz7cbGxp52m8g6deqwY8cOLrvssjTf/oXIkBf9SI2ITJkoNHo0Ll8+ug8eTOK0aQxdtozIFO5eIiISDJLe6vCbb77hgQceOGVe9uzZTxmfmJh42v2GA+XHH3+kdevW9OnThw8//JDIyEhWr17NV199lSbbO5csWbJQsGBBT7adGtozPhszGDCAvNdcw/A1a7i3TBmOHjrkdVUiIikqWLDgyUeePHlOmXfo0CHy5MnDmDFjaNCgAdmzZ+f999/no48+ImfOnKesJ6VDur/88gt169YlKiqKwoUL8+CDD7J3794z1jJp0iRq1arFM888Q9myZSlVqhQ333wzH3zwwSnjxo8fT4UKFciaNStFixalX79+Z73CV0p7vUkPZderV4/4+HiefPLJk0cEztTTubZ91VVX8f/+3/+jS5cu5M6dmyJFiqTZpTa1Z3wOFhFB3zlziLrpJp6ZNo2E4sUZs3o1WXPn9ro0EfFAvXr1Tpt3xx138NBDD5GQkEDTpk1PW96hQwc6dOjAX3/9RatWrU5b/uCDD9K0aVO2bt162h2SZsyYEajSAejVqxf9+/fngw8+IHPmzPzwww/nfM3y5ctp1KgRffr0YcSIEfzzzz/06NGDjh07nvEOUAULFmT16tUsXbqUSpUqpThm0aJFtG7dmmeffZa2bduyYMGCk8HXvXv3C+pv/PjxVKpUiY4dO/Lggw+ecVxqtz1gwAD69OnDk08+ybfffssjjzzCtddeS+3atS+ovjNRGKdSr+++I0fLljw6YQJ3ly7N+E2bINkhHxGRYNe9e/cU/yA4mzfeeIM777yTxx9//OS8oUOHUqVKFXbt2kWBAgVS3M7s2bOpXLkyRYoUoVatWtxwww3cc889J/fE33rrLerWrUufPn0AKF26NOvWreO111674DDOmzcvkZGR5MqV66yHpVO77UaNGp3c6+7evTuDBw/mxx9/VBh76ZHx44m6914u/+QTaNoUJk6EXLm8LktE0tHZ9lSjoqLOujxfvnxnXL5v3z6KFi0a8D3h5KpXr37er1m0aBHr16/ns88+OznvxOHcDRs2pBjGOXLkYPLkyWzYsIHp06czd+5cevXqxSuvvML8+fOJjo5m1apVNGvW7JTXXXvttfTp04e9e/eSOw2PQKZ22xUrVjxlTKFChdi1a1fA69F7xuep08cf0ywuDmbPZlKNGvwXH+91SSIiqZYjR45TpiMiIk57jzYxMfGU6ePHj9OpUyeWLFly8rF06VLWrVtH5cqVz7q9EiVK0KlTJ0aMGMFvv/3G9u3bGTp0KOAL9BPv6SZ3pvmpqTc1Urvt5Ce4mRnH0+DTNdozvhBt2vDnoUPcef/9lClXjmmLFpG/XDmvqxIROW/58+cnISHhlL3BJUuWnDKmatWqrFixgpIlS17UtmJjY4mKimL//v0AlC9fnjlz5pwyZs6cORQpUoRcZzjqmD9//lPOED906BCrV6+mSpUqJ+dlyZKFY8eOnbWWC9l2WtKe8QUq2LEj4/v2ZfXBg9SrUoUdyf7xioiEglq1apEjRw569erF+vXrGTduHO++++4pY55++mnmz59P165dWbx4MevXr+ebb76hS5cuZ1zviy++yFNPPcWMGTPYtGkTixcvpmPHjuzfv58WLVoA8PjjjzNz5kxefPFF1q5dS1xcHG+++SZPPfXUGdfboEED4uLimDFjBitWrKBjx46n7RnHxsYye/Zs/vjjjzNe5ONCtp2WFMYXofGzz/LtgAHEHz7MdTVqEP/zz16XJCJyXvLmzUtcXBzff/89FSpUYNiwYfTt2/eUMRUrVmTWrFls3ryZunXrUqlSJXr16kV0dPQZ11u3bl02bdrEvffeS7ly5bjpppvYvHkzEydO5Prrrwd8e9xffPEF48aN46qrrqJnz5707NnzrFfc6tWrFw0aNOCWW26hUaNGXHvttVStWvWUMS+99BJbt26lRIkS5M+fP8X1XMi205RzzpNHtWrVXCBNnz49oOs7H78OH+4uATcgTx7n1q276PV52UugqZfgEy59OJe2vaxcuTLN1p2SvXv3puv20lK49HKhfZzt3w6w0KWQidozDoCrO3VixdSpPBoZCdddx7Fly7wuSUREQojCOEAK33QTNmsWy48epXyVKiz+9FOvSxIRkRChMA6k8uXJPnYsh8yo37Ytvw4b5nVFIiISAhTGAVayYUNmzZpF/syZubFLF2YMHOh1SSIiEuQUxmkgpk4dZs2fT0zWrDR57DHmDRjgdUkicgGc7mUu5+lC/80ojNPI5ZUrM3PpUrrkz0+lp56C8eO9LklEzkPmzJk5ePCg12VIiDl48OAF3ZZSYZyG8pUpw8C1a8lWowb/tG7NxP/9z+uSRCSVChQowB9//EFCQoL2kOWcnHMkJCTwxx9/pHit7nPR5TDTWp48MG0aL5Urx+ABA3j/r7944JNPvK5KRM7hxKUht2/ffkHXPj5fhw4dIlu2bGm+nfQQLr2cbx+ZM2cmOjr6gm5woTBODzlz8srSpawtW5bOo0aRsH8/j+qwtUjQy507d5reOSipGTNmnHJ95VAWLr2kZx86TJ1OsufNy4T167mtUCF6TJjAy40aeV2SiIgECYVxOsqaOzefb9hA29hYRnz/PXsefxz0XpSISIanw9TpLFO2bHy8Zg1/dezIJW+9xbHERCIGDsQi9HeRiEhGpTD2QGSWLESPGsXxyy7jgcGDiZw2jfeWLSMySxavSxMREQ9od8wrZtiAARS65hpGrFlD+9KlOXrokNdViYiIBxTGHrKICP7fnDn0u/FGPo2P547ixTm8d6/XZYmISDpTGAeBZ6ZNY+BttzFhxw7ali5NxOHDXpckIiLpSGEcJB4dP57h7dvTZedOKvTsCfv2eV2SiIikE4VxEOn08cfcOHo0eZYtY1y1avwXH+91SSIikg4UxsGmbVt++t//aLtuHQ3KleOvNWu8rkhERNKYwjgIRTZrxlcvvcSqgwepW6kSO5Ys8bokERFJQwrjINX4uef4dsAA4g8f5vqaNdnyyy9elyQiImlEYRzE6vXowQ/Dh7M7MZEpzZvD+vVelyQiImlAYRzkru7UidXTptE1MhKuu46jS5d6XZKIiASYwjgEFLzxRpg5k0VHj1K2alUWjxnjdUkiIhJAqQpjM2tsZmvMbL2Z9Uxh+aVmNsHMlpnZfDO7KvClZnDly3PJ2LEkmlG/TRvmDh/udUUiIhIg5wxjM4sEhgBNgPLA3WZWPtmwZ4AlzrmKQHtgUKALFSjZsCGzZs4kf+bM3Ni5MzMGDvS6JBERCYDU7BnXBNY75zY6544AY4Fbko0pD/wI4JxbDcSaWXRAKxUAYq65hlnz51Msa1aaPPYYCwcP9rokERG5SKkJ48LA1iTT2/zzkloKtAQws5pADFAkEAXK6S6vXJkZixfTvUABKj3+OEyY4HVJIiJyEcw5d/YBZq2Bm5xznfzT7YCazrnuScbkxndougqwHCgLdHLOLU22rs5AZ4Do6OhqY8eODVgj+/fvJ2fOnAFbn5dS20um/fup0LMnh1au5LOWLanYrVs6VHd+MuLPJdiFSx+gXoJVuPSSFn3Ur19/kXOu+mkLnHNnfQC1ge+STPcCep1lvAGbgdxnW2+1atVcIE2fPj2g6/PSefWyb597sFAhZ+CGtWuXZjVdqAz7cwli4dKHc+olWIVLL2nRB7DQpZCJqTlMvQAoZWZXmFkW4C5gYtIBZpbHvwygEzDLOacb86aHnDl5c/lyGufPT+dRoxhUowbExkJEhO9rXJzXFYqIyDlkOtcA59xRM+sGfAdEAiOdcyvMrKt/+XtAOeATMzsGrATuT8OaJZnsefMyYf167o6JocfChRzAd3o78fHQubNvUNu2HlYoIiJnc84wBnDOTQGmJJv3XpLnvwKlAluanI+suXPzea5cdPjvPz4GHgFyAiQkQO/eCmMRkSCWqjCW0JBp2zY+Bv7FF8SH/POzbdniXVEiInJOuhxmOClWjEggn3+yE9AQ2F2woHc1iYjIOSmMw0m/fhAVdXLyVuA3oNbOnayaOPFMrxIREY8pjMNJ27YwbBjExIAZrWJimNm5MwlA7Vtu4cdXXvG6QhERSYHCONy0bQubN8Px47B5MzXff595c+ZQJFs27n7mGfa/9ZbXFYqISDIK4wwgpnZtfl6/ninXXkvOxx/HPfIIxxMTvS5LRET8FMYZxCWFC1N9xgx47DFef/ttWhUtyoE///S6LBERQWGcsURGwltvkb11a77auZO6xYuzfcECr6sSEcnwFMYZ0COff87Xffqw+uBBal19NUtHj/a6JBGRDE1hnEHd/PzzzP7yS5wZddu1458PP/S6JBGRDEthnIFVuf125v32G++XLk3ejh3h5ZfhHLfUFBGRwFMYZ3CFK1bkzqVLoU0bJvfuzaNXXsnRAwe8LktEJENRGAtkywajRzP3+usZvGoVLYoWZe+mTV5XJSKSYSiMxceMvjNn8t799zPt33+5tmxZtkyf7nVVIiIZgsJYTtFlxAimDBhAfGIitRo2ZMcXX3hdkohI2FMYy2ka9ejBL1On0umyyyh4990wYoTXJYmIhDWFsaToykaN6Lt+PdawIaseeIC36tfHHTvmdVkiImFJYSxndsklMHkyIypV4vEZM3igRAkS//vP66pERMKOwljOLlMm3li0iGcbNeKD+HgaFyvGvytWeF2ViEhYURjLOUVERtL3u+/4+LHHmL1vH7UrV2bbt996XZaISNhQGEuqtX/rLX4YOZIymTKR7/bb4euvvS5JRCQsKIzlvFx/3318vWkT2a66in9vvZUJHTroEpoiIhdJYSznr2BBmDGDV0qXpuXHH9OnenXckSNeVyUiErIUxnJhoqLou2QJ91aowIu//Ua7mBgO7djhdVUiIiFJYSwXLGv27Hy4dCn9br+duD//5IYSJdg9f77XZYmIhByFsVwUM+OZL7/ksxdeYPfhwxxt3Bhmz/a6LBGRkKIwloC448UX+f3337k8f36ONmjAb337el2SiEjIUBhLwGQuVw7mzuX1okWp9fzzjGzeHI4f97osEZGgpzCWwLr0Uh6eN4/6hQpx/+TJjOvQgeMHDnhdlYhIUFMYS8Bdkj8/kzdtokutWryzdSt3FC1KwsaNXpclIhK0FMaSJjJnycLQX3/lySZN+O7ff1l3zTWwfLnXZYmIBCWFsaQZM6PpU0+x4fvvqRQRAXXqsDMuzuuyRESCjsJY0lyBG26A+fP5OG9eSt1zD1MffNDrkkREgorCWNJH4cI0nDaNErlz0+y99xhSty4cPep1VSIiQUFhLOmmSJkyzN6yhWbFi9Nt1ix6lCzJsX//9bosERHPKYwlXeW85BImrF3LYzfcwKD4eGZWqQKbN3tdloiIpxTGku4iIyN56/vvWfjuuzT47z+oVYsjuoSmiGRgCmPxTLUHH4S5c5kdGUmZ669n0SuveF2SiIgnFMbirbJlufSzzzieNSvXP/MMX7VpA855XZWISLpSGIvnrrruOuatWcNVl11GyzFjeLN6ddyhQ16XJSKSbhTGEhQKxsQwY8sWWl15JU/89hsTqlSB3bu9LktEJF0ojCVoZI+KYuyyZYzu1o1bN22Cq6+GVau8LktEJM0pjCWoRERE0Pbtt4mYOZP4PXtoWKECm0aN8rosEZE0pTCW4FSrFn8MG8ZioFb79vz69NNeVyQikmYUxhK06rRsya/z5pE7Kor6r7/O2GbN4Ngxr8sSEQk4hbEEtTLVqjF3wwZqFirE3VOm8FmtWrB/v9dliYgElMJYgl6+ggX5fuNGnmvcmKa//QbXXQfbtnldlohIwCiMJSRkzZqVl779llyTJ3Ng/XoeKlOGv3/80euyREQCQmEsoaVJExYOHszIhASuvvFG1g4Z4nVFIiIXTWEsIafufffx06RJ/BcZydXdujGjSxddQlNEQprCWEJSnebNmbd0KQVz56bRsGGMu+EGOHLE67JERC5IqsLYzBqb2RozW29mPVNYfomZTTKzpWa2wszuC3ypIqcqXr48v2zaxO3ly1Plp5+gcWP45x+vyxIROW/nDGMziwSGAE2A8sDdZlY+2bCHgZXOuUpAPeBNM8sS4FpFTpMnb17GrFhB8U8+4ficObxTtiwHly/3uiwRkfOSmj3jmsB659xG59wRYCxwS7IxDshlZgbkBP4Bjga0UpGzadeOXwcO5JHdu6lfpQo7J0zwuiIRkVQzd44TX8ysFdDYOdfJP90OqOWc65ZkTC5gIlAWyAXc6ZybnMK6OgOdAaKjo6uNHTs2UH2wf/9+cubMGbD1eUm9XLi5X33FS4MHU8A5BnfsSO527QK27nD5uYRLH6BeglW49JIWfdSvX3+Rc676aQucc2d9AK2BEUmm2wFvJxvTChgAGFAS2ATkPtt6q1Wr5gJp+vTpAV2fl9TLxVk4fbq7PEsWlxvctKuvdq5YMefMnIuJcW706Ateb7j8XMKlD+fUS7AKl17Sog9goUshE1NzmHobUDTJdBFge7Ix9wHj/dta7w/jsqn+U0EkgKrVq8e8VasonSsXmebOhS1bfB99io+Hzp0hLs7rEkVETpGaMF4AlDKzK/wnZd2F75B0UluAhgBmFg2UATYGslCR81G0eHHm5clDff/0JOAwQEIC9O7tXWEiIik4Zxg7544C3YDvgFXA5865FWbW1cy6+of1BeqY2XLgR+Bp59xfaVW0SGpE+K9fvRrfGYfX4jtkw5Yt3hUlIpKCTKkZ5JybAkxJNu+9JM+3A40CW5rIRSpWDOLjKQuMBzoAVYGPs2WjxZEjkEWfvhOR4KArcEn46tcPoqIAuBX4DSgO3HLwIL1jY33vIYuIBAGFsYSvtm1h2DCIiQEzisfE8PPIkTx0000U/vdfqFIFJp/2CTwRkXSnMJbw1rYtbN4Mx4/D5s1ku+8+hkydykPLl0NMDBOaN2faXXfBUV2jRkS8ozCWjKlkSdzPP9M/OprGn33GCyVLcsx/wpeISHpTGEuGZVFRfL9xI/dedx0vxcdzU/Hi7PzyS6/LEpEMSGEsGVpUVBQfzprFyL59+fnoUaq0bs3uXr18h7VFRNKJwlgEuO/ZZ5n3yy88WKEC+V99FZo1g7/0UXkRSR8KYxG/ildfzXNLl8J777Hkhx9oVawYf0+d6nVZIpIBKIxFkjKDLl1Y+eKLTDp4kKpNmjCvRw/fta1FRNKIwlgkBW169+bnn34iIiqK6wYNYlDlykTs2+d1WSISphTGImdQvX59ftu6lSZXXkmPZctY2749LF7sdVkiEoYUxiJncWnevHy1fDmjnn+euyIjoXZtEocO1WFrEQkohbHIOZgZ9/Tpw9IRI9hdpw7lH3qI4bVr43TYWkQCRGEskkqJefJgY8dyRYkSdJ43j3uLFuXAwoVelyUiYUBhLHIe8hUowLdr1tDn3nsZvWcPNWvWZOXrr3tdloiEOIWxyHmKjIzk+Y8+4vuxY/krUyaeffpp6NIFDh3yujQRCVEKY5EL1PDOO1m8fj3vd+8Ow4axu0YNDq1c6XVZIhKCFMYiF6FQsWLkHzwYN3Eit69eTe0KFVj/7rtelyUiIUZhLBIAdvPNPP3++8SbUe3hhxl3882QmOh1WSISIhTGIgHSrGNHFq9aRbnoaFp98w09ihXjyIYNXpclIiFAYSwSQDGlSjFryxYeadKEb3bu5GCNGqCbTYjIOSiMRQIsS5YsDJoyhd8WLuSSIkU41KQJP91zDxw75nVpIhKkFMYiaSR31aowdy79q1alYVwczxQvztFt27wuS0SCkMJYJC1FRfH4nDk8ULcur2zZQsPixdk+bpzXVYlIkFEYi6Sx7NmzM2zGDD7p14+FR49SpVUrfu7aFY4f97o0EQkSCmORdNLumWeYP3cusXnzku/996FFC/jnH6/LEpEgoDAWSUdX1qzJ3N27KfPOO7jvvqN/iRLs/u47r8sSEY8pjEXSmUVEwMMPszoujmf/+48qjRvz82OP6R7JIhmYwljEI+XuuINfZ8wgW1QUdQcO5M3KlXF79nhdloh4QGEs4qEqdeuyaNs2brnqKp5Ytoz7ixWDpUu9LktE0pnCWMRjl1x6KV8uW8bA7t1plSkTXH01jBzpdVkiko4UxiJBwMx4dPBgmq5aBddcw1v338+7tWvjDhzwujQRSQcKY5FgUqAAbupUZpYpw8Nz59KmcGH2LVrkdVUiksYUxiJBxjJlYsLKlbx83318vmcP1WvUYPkbb3hdloikIYWxSBCKiIig18iR/PT55+zNlIk6Tz3FXx07wuHDXpcmImlAYSwSxOq2bs3iDRsY3rw5+T78EK69lmPr13tdlogEmMJYJMgVLFqUuyZNgq++4tuVK6lapgxr3n3X67JEJIAUxiKh4pZbyPzee2yPiKD6ww/zWYsWkJjodVUiEgAKY5EQckO7dixetYqKBQty16RJPBwTw+GNG70uS0QuksJYJMQUKVmSGVu28HjTpry7YwfjK1eGH37wuiwRuQgKY5EQlDlzZvpPnsy8zz7jrqJFoVEjdj75JBw75nVpInIBFMYiIazmHXdg8+ez4dZbKd2/P0+WKEHi9u1elyUi50lhLBLqcuSgcFwc99SvT//4eOpfcQXbxo/3uioROQ8KY5EwkC17dob89BNjXn2VpYmJVLn9dqY98IDukSwSIhTGImHkrqefZuH8+RTMnZvxI0bArbfCv/96XZaInIPCWCTMlKlenXnbtzOwf3/49ltWVajAzmnTvC5LRM5CYSwShqJy5CDb449zfOZM7tq1iyo33cSsJk0gJoa6DRpAbCzExXldpoj4KYxFwlhE7dqM/vFHcmXNSv2pU3l1yxaccxAfD507K5BFgoTCWCTMVbjuOhbmz09roBdwM7AXICEBevf2tDYR8VEYi2QAuf74gzHAO0ACkOPEgi1bPKtJRP5PqsLYzBqb2RozW29mPVNY/qSZLfE/fjezY2aWN/DlisgFKVYMAx4GfgQigd1AFzP+nj7d09JEJBVhbGaRwBCgCVAeuNvMyicd45x7wzlX2TlXGd+RsJnOuX/SoF4RuRD9+kFUFPB/v/RzMmXiw+PHuapBAya1aaM7QIl4KDV7xjWB9c65jc65I8BY4JazjL8bGBOI4kQkQNq2hWHDICYGZwYxMdz20UcsmDmTAnny0GLMGDpER/PfnDleVyqSIZk7xxV6zKwV0Ng518k/3Q6o5ZzrlsLYKGAbUDKlPWMz6wx0BoiOjq42duzYi+/Ab//+/eTMmTNg6/OSeglO4dJL8j4SExP5sl8/Ppg5k45m9OzYka13342LjPSwytQJl58JqJdglBZ91K9ff5FzrvppC5xzZ30ArYERSabbAW+fYeydwKRzrdM5R7Vq1VwgTZ8+PaDr85J6CU7h0suZ+pg/bZr789ZbnQO3tWJFt3fevPQt7AKEy8/EOfUSjNKiD2ChSyETU3OYehtQNMl0EeBMt4W5Cx2iFglJNW68kegJE3Cff06blSupWKsWMzp31m0ZRdJBasJ4AVDKzK4wsyz4Andi8kFmdglQF/g6sCWKSHqy1q159auvyJQjB/WHD+eRIkVIWLLE67JEwto5w9g5dxToBnwHrAI+d86tMLOuZtY1ydDbgGnOuQNpU6qIpJc6zZqxdOdOHmncmLf//JNKVauytlcv7SWLpJFUfc7YOTfFOVfaOVfCOdfPP+8959x7ScZ85Jy7K60KFZH0FZUjB4O+/ZafvvyS2Msuo/Crr0LdurBundeliYQdXYFLRM6q/u238/2uXeQYNYqE33/ntrJlWfTkk3D8uNeliYQNhbGInJsZ3HMPG8ePZ0GWLNTq358XihfnyOrVXlcmEhYUxiKSalc1aMDyP/6gTe3avBQfT60rr2RZ797aSxa5SApjETkvl+bNyye//MJXI0awPVMmur/8MjRq5Lsto4hcEIWxiFyQW+6/nxXbtvHxyy/DvHnsvvJKVr30Epzjqn4icjqFsYhcsHz58xPbqxcsX07P3Lmp8sILvFmuHMc2b/a6NJGQojAWkYsXG0u/hQtpXLEiT6xZQ72SJVn/2mvaSxZJJYWxiAREwUKFmLBkCZ/0789yoFLPnnxbqxZsP9PVc0XkBIWxiASMmdHu8cf5feNGbqlalWrLl8OVV+JGjdJesshZKIxFJOCKFCvGp4sWUWDZMo6VK0ez9u0ZUbUq7s8/vS5NJCgpjEUk7ZQqxZ6vv+ZQiRI8sGQJzYoV44+hQ7WXLJKMwlhE0lTe/Pn5Ye1a3n72WWYeO8ZVDz3EqFq1cLt2eV2aSNBQGItImouIiKBb374sXbGCK2Ni6LNgAYeuvBLGjfO6NJGgoDAWkXRTsmxZZm7YwPTvvyd7TAwJrVox8frr4e+/vS5NxFMKYxFJV5GRkRS94Qb49VeGNm7MLbNnc1fhwvw1apTXpYl4RmEsIt7InJlHJ02iX7dujD9yhKvat2di/frw779eVyaS7hTGIuKZTJky8czbb7Ng/nwKRkdzy4wZvBoTA5Mne12aSLpSGIuI5ypVr878LVt4oXNnbitYEJo352iHDrBnj9eliaQLhbGIBIUsWbLw4vvvU2b5clyvXrT5+GO6FC7MvgkTvC5NJM0pjEUkuGTNyvG+fYm95x6GHzhAxZYtmdG8Oezb53VlImlGYSwiQScyMpLXR41i9o8/kilPHupPnswjhQuToPeSJUwpjEUkaF3ToAFLtm2je6tWjE1IYF/z5vDww7B/v9eliQSUwlhEglqOHDkY/MUXrNmyhejHHuPYu+/yTmwsh6ZN87o0kYBRGItISLi0UCF46y1+7N+f7n//TbWbbuLASy9BQoLXpYlcNIWxiISURo8/zrcTJrAnRw5umT6d54sU4ciMGV6XJXJRFMYiEnIa33orv2/bRpPq1en777+0qV8fnngCDh70ujSRC6IwFpGQlCdPHh5/4w2+HjuW/916K7z5JocqVeLozz97XZrIeVMYi0hIa3HnndSZMAGmTaPnjh3UufZaVnXqBIcOeV2aSKopjEUkPNx4I9e88w4bs2alygcf0D8mhmPz5nldlUiqKIxFJGy0vvdeVsTH06R2bZ7ctYu6V1/Nxm7d4MgRr0sTOSuFsYiElejoaMb//DOj3nuPTVFRJA4ZAjVqwOLFXpcmckYKYxEJO2bGPV26sPGffygzcSJu5076Va/O5h49IDHR6/JETqMwFpGwlTVrVrj5ZjZ/+y2vRURQYdAghpcogVu6FOLiIDYWIiJ8X+PivC5XMjCFsYiEvSuqVGH5+vXUrFCBzlu30rRyZbZ16ADx8eCc72vnzgpk8YzCWEQyhJiYGL5fsoR3XnmFWUCDo0c5lnRAQgL07u1RdZLRKYxFJMOIiIjg4Z49WQq8B0QCR4H4EwO2bPGqNMngFMYikuGUjImhgf/5EKAs0Bc4lDev77C1SDpTGItIxtOvH0RFAdASuBl4Hqjw999MrV4d1q3zsjrJgBTGIpLxtG0Lw4ZBTAxFzfg8JoZpTz1FRP78NPntN54uWxaef143npB0ozAWkYypbVvYvBmOH4fNm7nxtddYtnUrr/TqRfMGDaBvXw6UK8fhCRO8rlQyAIWxiIhf1qxZ6fnyy1z3/ffw0088s3cvFVu25PvatX3BLZJGFMYiIimpX5+mn3zC8csuo9HcubQuUYKtTz4Jhw97XZmEIYWxiMgZ3NS8Ocu3baPvE0/wDVC2f3/GXXEFfP+916VJmFEYi4icRbZs2Xj2jTdYtWEDTa+9lkqZM0OjRhxp1Qq2bfO6PAkTCmMRkVSIjY3li9mzKblmDbz0Eq0mTODu2Fj+ePZZ3XxCLprCWETkfGTLxvHevanWvTsTnKNsv368VbQoiT/+6HVlEsIUxiIi5ykiIoIXBg5k5bp11K1encd37qTKDTew6uab4c8/vS5PQpDCWETkAhUvXpxJ8+fz9eefk6NQIQp+9x2UKYMbNAiOHvW6PAkhCmMRkYtgZrRo3Zq527Zx6e+/c7RmTer16MGg2FiOzp7tdXkSIhTGIiIBYGZQujR7xowhe6VK9PjjD6pdfz1zmjWD3bu9Lk+CXKrC2Mwam9kaM1tvZj3PMKaemS0xsxVmNjOwZYqIhIbL8uXj28WLGTd6NP/mysV1U6Zwb5Ei7B04EI4dO+frJWM6ZxibWSS+u4w1AcoDd5tZ+WRj8gDvAi2cc1cCrQNfqohIaDAzWrZty6odO+jVqRPLs2Qh+2OPQe3asHCh1+VJEErNnnFNYL1zbqNz7ggwFrgl2Zg2wHjn3BYA59yuwJYpIhJ6cuTIwcvDhzP/n3/IPHo0ezdvpkmNGvzasiX8+6/X5UkQSU0YFwa2Jpne5p+XVGngUjObYWaLzKx9oAoUEQl1mTJnhrZt2fDllyzPkYM6EyZw/+WXs3vQIN9doyTDM+fc2QeYtQZucs518k+3A2o657onGfMOUB1oCGQHfgWaOefWJltXZ6AzQHR0dLWxY8cGrJH9+/eTM2fOgK3PS+olOIVLL+HSB4RmLwcPHmTsoEF8+t135AKeK1SIOi+8wO5ChUKulzMJxZ9LStKij/r16y9yzlU/bYFz7qwPoDbwXZLpXkCvZGN6Ai8mmf4AaH229VarVs0F0vTp0wO6Pi+pl+AULr2ESx/OhXYvK5Yvd/XLlnU3Zc7sjkdEuK233+7cnj1elxUQofxzSSot+gAWuhQyMTWHqRcApczsCjPLAtwFTEw25mvgOjPLZGZRQC1g1fn/zSAikjGUv+oqfly5ks/XrsU6dyZx3DgeKliQv99/H85xxFLCzznD2Dl3FOgGfIcvYD93zq0ws65m1tU/ZhUwFVgGzAdGOOd+T7uyRURCn5mROzYWhg7lk3vvZdjBg5Tu2pVh5cpx/Hf9F5qRpOpzxs65Kc650s65Es65fv557znn3ksy5g3nXHnn3FXOuYFpVK+ISFiq26EDS5Ys4apSpeiyZg1XV6jAwvbtYf9+r0uTdKArcImIBImrKlVixpo1jBoyhC3ZszNi1CgoVw7GjdOh6zCnMBYRCSJmxj0PPcSaHTt49dtvIW9e5rdqxQcVKnB8zRqvy5M0ojAWEQlCl1xyCXkaN4ZFi/iwTh06rVjBteXKsfiBB+DgQa/LkwBTGIuIBLNMmXh3zhw+HjSIDVmzUn3ECLpHR/PfmDFeVyYBpDAWEQlyZkb7Rx5hzY4dPHTrrby7bx8j27SBFi1g0yavy5MAUBiLiISIPHny8PaECSxeuJBuL78MP/3Ej2XKsKxbNzh82Ovy5CIojEVEQkzFatXI0qsXbuVKnsiRg6pDhtCjYEH2TJjgdWlygRTGIiIhyooV48cNG3igWTMG//cfZVq2ZHTNmritW8/9YgkqCmMRkRCWN29ehn7zDfN//plihQvTbsECvi9VCt54AxITvS5PUklhLCISBqrXqcPcLVv46v33ufHGG+Gpp/ipVCn2TZnidWmSCgpjEZEwERERwS2dO2OTJvHfp59yy5YtlG3WjLHXXovbscPr8uQsFMYiImEoz91388P06Vx++eXc/fPPNCxalJW9esHRo16XJilQGIuIhKladesyb+tWhvbpwxIzqrz6Kn9UqgS//OJ1aZKMwlhEJIxFRkbS9fnnWfPHHwzv1o3Ce/fCNdfwW4sWuF27vC5P/BTGIiIZQP4CBWj/9tuwahVL7r2X6pMmcVPhwqx58UU4dgzi4iA2FiIifF/j4jyuOGNRGIuIZCQ5c1Lhgw8Y/MwzzHeOCn368Mxll3Hg/vshPt53q8b4eOjcWYGcjhTGIiIZTGRkJN369WPNtm20ue46Xtmzh+qHD3PKqV0JCdC7t1clZjgKYxGRDCq6YEE+mjWLWcDTQCbAActODNiyxavSMhyFsYhIBnddTAwd/M8nApWAVsCaXLngyBHP6spIFMYiIhldv34QFQVAA6AP8B1w5d69dMmXj+1DhsDx415WGPYUxiIiGV3btjBsGMTEkMuM52Ni2DBkCA+3aMGH+/fToFs3XJUqMGWK7wQvCTiFsYiI+AJ582bfHvDmzRR46CEGff01q9eu5f3nnsP27+dws2a8W7o0h6ZP97rasKMwFhGRMypesiR1X3oJVq3im06deHj9eko3aMBHVapwbNmyc69AUkVhLCIi55YlC7cPH85PkydTsHBh7luyhEqVKjGxQQPc5s1eVxfyFMYiIpJq9Zs2Zd7WrXwxciSJl15KvxkzoHRpeOwx2L3b6/JClsJYRETOi5nR6r77+H3nTsbNnYu1b8/uQYO4s1Ahfn/oISITErwuMeQojEVE5IJkzpyZIjVrwogRLB0+nKlApaFDGXHbbWx58UU4fNjrEkOGwlhERC7aDfffz8Y//6RHmzaMTUykdJ8+PFmgAMc//th3Iwo5K4WxiIgExGWXXcabcXF8HBfHXTfeyNaICCI6dIDKlTn69df6jPJZKIxFRCSgCl5+OR9Nm0bcrl3w2Wes3LuX4rfeyrBSpTg6Y4bX5QUlhbGIiKSJyMyZ4Y47ODp+PEWLF6fLhg1cVb8+46pXxy1d6nV5QUVhLCIiaapitWrMWb+erz//nMgCBWi1aBH1K1fm+D33wKZNXpcXFBTGIiKS5syMFq1bs2z7dkYOHkyzevWIGDcOV7o069q3h507vS7RUwpjERFJN5GRkdzXvTtPTp8O69fzfaNGlBk1inaFC7PpkUdg716vS/SEwlhERLxRuDA14+J4+oEHGAeUefttHo2OZnffvnDokNfVpSuFsYiIeCZPnjy8MmwY6+Lj6XDrrQw5fJg6zz/P8dKl4aOPMsxnlBXGIiLiucKFCzNswgR+X7mSt19+mYjoaI7edx8fFivGkS+/DPvPKCuMRUQkaJQtW5bGvXrB/PlMeuIJOm7fTrnWrRlTpgzHw/gzygpjEREJPmbc+vrrTJ44kRyFC9Nm3Tqq16/PtBo1cIsXe11dwCmMRUQkKJkZTW++mcXx8XwyYgT/XHopTyxejKtaFdq0gQ0bvC4xYBTGIiIS1CIjI2l3//2s2bGDrxYuJKJXL/6bMIGOpUqxrm1b+PNPr0u8aApjEREJCVmzZqV45crw8sss/OgjPo+MpPynn/Jg0aLsePRR2LPH6xIvmMJYRERCzg133smGbdvo0rYtI44do+TgwTxbsCDH3ngjJD+jrDAWEZGQFB0dzTujR7Nq7Vpa3HQTS3PlIvKpp6BUKY4PHw5Hj3pdYqopjEVEJKSVLFmSMVOnMv6PP+Cnn9iYNy9lO3dmVLFiHPvii5D4jLLCWEREwkLmzJmhfn32fvQRuUuUoP2OHVS54w6mlC2L++knr8s7K4WxiIiElcpVqjB/7VrGxsWRUKAAzdaupVHDhhxr1Ah++83r8lKkMBYRkbATERHBnW3asGrbNoYMHEidG28kctEiqFaNrS1awLp1Xpd4CoWxiIiErcyZM/PQo4/SZ9o02LiROe3bc8WkSXQqU4Zt99wD27d7XSKgMBYRkYzikkso078/3R94gE/MKBUXx9MxMfz72GPw33+elqYwFhGRDCN//vwMGDaMtRs20PrWW3nj6FGqDxzIsSuugNdfh4QEiIuD2FjqNmgAsbG+6TSWqjA2s8ZmtsbM1ptZzxSW1zOzPWa2xP94PvClioiIBEZsbCyfTJjAkqVLeaN/fyLr1OH400/zef78HL3vPoiPx5yD+Hjo3DnNA/mcYWxmkcAQoAlQHrjbzMqnMHS2c66y//FSgOsUEREJuIoVK9Ly8cdh8mQmv/wydyYkUDExka+Ak59OTkiA3r3TtI7U7BnXBNY75zY6544AY4Fb0rQqERGRdNa8Z0/GA8eB24ABSRdu2ZKm2zZ3jiuTmFkroLFzrpN/uh1QyznXLcmYesA4YBuwHXjCObcihXV1BjoDREdHVxs7dmxgugD2799Pzpw5A7Y+L6mX4BQuvYRLH6BeglUo93L1XXeRaedOPgRuBfL75x+KjmZuADKrfv36i5xz1U9b4Jw76wNoDYxIMt0OeDvZmNxATv/zpsC6c623WrVqLpCmT58e0PV5Sb0Ep3DpJVz6cE69BKuQ7mX0aOeiopzzXUTT94iK8s0PAGChSyETU3OYehtQNMl0EXx7v0kDfa9zbr//+RQgs5nlO68/F0RERLzWti0MGwYxMTgziInxTbdtm6abTU0YLwBKmdkVZpYFuAuYmHSAmRU0M/M/r+lf79+BLlZERCTNtW0Lmzcz86efYPPmNA9igEznGuCcO2pm3YDvgEhgpHNuhZl19S9/D2gFPGhmR4GDwF3+3XERERE5h3OGMZw89Dwl2bz3kjx/B3gnsKWJiIhkDLoCl4iIiMcUxiIiIh5TGIuIiHhMYSwiIuIxhbGIiIjHFMYiIiIeUxiLiIh4TGEsIiLiMYWxiIiIx855C8U027DZbiA+gKvMB/wVwPV5Sb0Ep3DpJVz6APUSrMKll7ToI8Y5lz/5TM/CONDMbKFL6R6RIUi9BKdw6SVc+gD1EqzCpZf07EOHqUVERDymMBYREfFYOIXxMK8LCCD1EpzCpZdw6QPUS7AKl17SrY+wec9YREQkVIXTnrGIiEhICoswNrPGZrbGzNabWU+v6zkbMytqZtPNbJWZrTCzR/3z85rZ92a2zv/10iSv6eXvbY2Z3eRd9Skzs0gzW2xm3/inQ7IXM8tjZl+a2Wr/z6d2KPZiZo/5/239bmZjzCxbKPVhZiPNbJeZ/Z5k3nnXb2bVzGy5f9lgM7Mg6OMN/7+vZWY2wczyBHsf/hpO6yXJsifMzJlZviTzQq4XM+vur3eFmb2eZH769OKcC+kHEAlsAIoDWYClQHmv6zpLvZcDVf3PcwFrgfLA60BP//yewGv+5+X9PWUFrvD3Gul1H8l6+h/wKfCNfzokewE+Bjr5n2cB8oRaL0BhYBOQ3T/9OdAhlPoArgeqAr8nmXfe9QPzgdqAAd8CTYKgj0ZAJv/z10KhjzP14p9fFPgO3zUj8oVqL0B94Acgq3+6QHr3Eg57xjWB9c65jc65I8BY4BaPazoj59wO59xv/uf7gFX4/gO9BV8Y4P96q//5LcBY59xh59wmYD2+noOCmRUBmgEjkswOuV7MLDe+X9IPAJxzR5xz/xGCvQCZgOxmlgmIArYTQn0452YB/ySbfV71m9nlQG7n3K/O9z/nJ0leky5S6sM5N805d9Q/ORco4n8etH34607pZwIwAHgKSHryUSj28iDwqnPusH/MLv/8dOslHMK4MLA1yfQ2/7ygZ2axQBVgHhDtnNsBvsAGCviHBXt/A/H9Mh5PMi8UeykO7AY+9B9yH2FmOQixXpxzfwD9gS3ADmCPc24aIdZHCs63/sL+58nnB5OO+PaoIAT7MLMWwB/OuaXJFoVcL0Bp4Dozm2dmM82shn9+uvUSDmGc0nH6oD9F3MxyAuOAHs65vWcbmsK8oOjPzJoDu5xzi1L7khTmBUUv+PYmqwJDnXNVgAP4DoeeSVD24n8v9RZ8h9QKATnM7J6zvSSFeZ73cR7OVH9Q92VmvYGjQNyJWSkMC9o+zCwK6A08n9LiFOYFbS9+mYBLgauBJ4HP/e8Bp1sv4RDG2/C9b3FCEXyH5YKWmWXGF8Rxzrnx/tk7/Yc+8H89cZgkmPu7BmhhZpvxvT3QwMxGE5q9bAO2Oefm+ae/xBfOodbLDcAm59xu51wiMB6oQ+j1kdz51r+N/zsEnHS+58zsXqA50NZ/iBNCr48S+P7gW+r//S8C/GZmBQm9XsBX23jnMx/fkb58pGMv4RDGC4BSZnaFmWUB7gImelzTGfn/2voAWOWceyvJoonAvf7n9wJfJ5l/l5llNbMrgFL4ThzwnHOul3OuiHMuFt/3/Sfn3D2EZi9/AlvNrIx/VkNgJaHXyxbgajOL8v9ba4jvvIRQ6yO586rffyh7n5ld7f8+tE/yGs+YWWPgaaCFcy4hyaKQ6sM5t9w5V8A5F+v//d+G78TUPwmxXvy+AhoAmFlpfCdw/kV69pIWZ6ul9wNoiu+s5A1Ab6/rOUet1+I7nLEMWOJ/NAUuA34E1vm/5k3ymt7+3tbgwdmHqeyrHv93NnVI9gJUBhb6fzZf4TtsFXK9AH2A1cDvwCh8Z4KGTB/AGHzvdyfi+0/+/gupH6ju/x5sAN7Bf5Ejj/tYj+89yBO/++8Fex9n6iXZ8s34z6YOxV7whe9of22/AQ3SuxddgUtERMRj4XCYWkREJKQpjEVERDymMBYREfGYwlhERMRjCmMRERGPKYxFREQ8pjAWERHxmMJYJIMws7fNbFIK83Ob2YtmVi7JvMfMd89d/R8hkg70iyaSAZhZCaALvqtzJVcdeAHInGTee/jujHRvCuNFJMAUxiIZQw9gqXNuYQrLqgCH8V2LGwDn3EF892h9Il2qE8ngFMYiYcLMWvvvxbrXzPab2QIza2ZmWYF7gE9TeM0qfPc/zgokmpkzsy/9i8cC5c2sTro1IZJBKYxFwoCZ9ccXnivwBe/dwC9Abnz3aM0DzE7hpe2BjcAkoLb/8bh/2RJgL9A47SoXEfDdUFlEQpiZ3YUvQFs7575MsmiSf/nT/N+dwpJbiu9erG875+YmXeCcO25my/CFuYikIe0Zi4S+PsCkZEGcVCFgr3PuSArLrsR3+7jfzvDa3f7Xi0gaUhiLhDAzKw6UBr44y7Bs+E7QSklVfHvNS86w/CCQ/ULrE5HUURiLhLYTe63bzzLmb+DSMyyrAmxwzu09w/K8wF8XWJuIpJLCWCS0nQjhK88yZjWQ2cyKpLCsPEk+0pSCK4A1F1ibiKSSTuASCW2bgDlAHzMD30la0UBToJ9zbi0wyz+2JrAt2ev/A6qa2U3AHmCdc+5vADPLg+8QeP+0bUFEtGcsEsKccw64HfgS3wU6vgPewPe7vck/ZjMwH7g5hVU8D+wEvgJ+BcolWdYMOAJMSJPiReQk8/0ui0g4M7MOwCDgcudcQipf8y3wl3OuXVrWJiIKY5EMwcwigeXASOfcOQ87m1llYC5wlXNufRqXJ5Lh6TC1SAbgnDsGdARStVcMFATuUxCLpA/tGYuIiHhMe8YiIiIeUxiLiIh4TGEsIiLiMYWxiIiIxxTGIiIiHlMYi4iIeExhLCIi4rH/D0uaJksKJEg2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.plot(t_leapfrog, u_leapfrog, 'or-', label=\"Leap-Frog\")\n",
    "axes.plot(t_exact, u_exact, 'k--', label=\"True Solution\")\n",
    "axes.grid()\n",
    "axes.set_title(\"Leap-Frog\", fontsize=18)\n",
    "axes.set_xlabel(\"t (years)\", fontsize=16)\n",
    "axes.set_xlabel(\"$c(t)$\", fontsize=16)\n",
    "axes.legend(loc='best', fontsize=14)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Error Analysis of Leap-Frog Method\n",
    "\n",
    "To easily analyze this method we will expand the Taylor series around time $t_{n+1}$ to yield\n",
    "$$\\begin{aligned}\n",
    "    u_{n+2} &= u_{n+1} + \\Delta t f(t_{n+1},u_{n+1}) + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  + \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We need one more expansion however due to leap-frog.  Recall that leap-frog has the form\n",
    "$$\n",
    "    U_{n+2} = U_{n} + 2 \\Delta t f(t_{n+1}, U_{n+1}).\n",
    "$$\n",
    "To handle the $U_{n}$ term we need to write this with relation to $u(t_{n+1})$.  Again we use the Taylor series\n",
    "$$\n",
    "    u(t_n) = u_{n+1} - \\Delta t f_{n+1} + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  - \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\\begin{aligned}\n",
    "    u(t_{n+2}) &= u_{n+1} + \\Delta t f_{n+1} + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  + \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4)\\\\\n",
    "    u(t_{n}) &= u_{n+1} - \\Delta t f_{n+1} + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  - \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4)\n",
    "\\end{aligned}$$\n",
    "\n",
    "Plugging these into our definition of the truncation error along with the leap-frog method definition leads to\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    T(t, u; \\Delta t) &= \\frac{1}{\\Delta t} \\left [\\underbrace{U_{n} + 2 \\Delta t f_{n+1}}_{U_{n+2}} - \\underbrace{\\left(u_{n+1} + \\Delta t f_{n+1} + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  + \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4) \\right )}_{u(t_{n+2})} \\right ] \\\\\n",
    "    &=\\frac{1}{\\Delta t} \\left [ \\underbrace{ \\left(u_{n+1} - \\Delta t f_{n+1} + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  - \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4)\\right)}_{u_{n}} + 2\\Delta t f_n - \\underbrace{\\left(u_{n+1} + \\Delta t f_{n+1} + \\Delta t^2 \\frac{u''(t_{n+1})}{2}  + \\Delta t^3 \\frac{u'''(t_{n+1})}{6} + \\mathcal{O}(\\Delta t^4) \\right )}_{u(t_{n+2})} \\right ] \\\\\n",
    "    &=\\frac{1}{\\Delta t} \\left [- \\Delta t^3 \\frac{u'''(t_n)}{3} + \\mathcal{O}(\\Delta t^4) \\right ] \\\\ \n",
    "        &=- \\Delta t^2 \\frac{u'''(t_n)}{3} + \\mathcal{O}(\\Delta t^3)\n",
    "\\end{aligned}$$\n",
    "\n",
    "Therefore the method is second order accurate and is consistent theoretically.  In practice it's a bit more complicated than that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Compare accuracy between Euler, RK2 and Leap-Frog\n",
    "f = lambda t, u: -u\n",
    "u_exact = lambda t: numpy.exp(-t)\n",
    "u_0 = 1.0\n",
    "\n",
    "t_span = (0.0, 10.0)\n",
    "num_steps = [2**n for n in range(4,11)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "delta_t = numpy.empty(len(num_steps))\n",
    "error_euler = numpy.empty(len(num_steps))\n",
    "error_RK2 = numpy.empty(len(num_steps))\n",
    "error_leapfrog = numpy.empty(len(num_steps))\n",
    "\n",
    "for (i, N) in enumerate(num_steps):\n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    tt, u_euler = euler(f, t_span, u_0, N )\n",
    "    tt, u_rk2 = RK2(f, t_span, u_0, N)\n",
    "    tt, u_leapfrog = leap_frog(f, t_span, u_0, N, start=euler)\n",
    "    \n",
    "    delta_t[i] = t[1] - t[0]\n",
    "        \n",
    "    # Compute error for each\n",
    "    error_euler[i] = numpy.linalg.norm(delta_t[i] * (u_euler - u_exact(t)), ord=1)\n",
    "    error_RK2[i] = numpy.linalg.norm(delta_t[i] * (u_rk2 - u_exact(t)), ord=1)\n",
    "    error_leapfrog[i] = numpy.linalg.norm(delta_t[i] * (u_leapfrog - u_exact(t)), ord=1)\n",
    "    \n",
    "# Plot error vs. delta_t\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "order_C = lambda delta_x, error, order: numpy.exp(numpy.log(error) - order * numpy.log(delta_x))\n",
    "axes.loglog(delta_t, error_euler, 'bo', label='Forward Euler, $n=1$')\n",
    "axes.loglog(delta_t, error_RK2, 'ro', label='RK2, $n=2$')\n",
    "axes.loglog(delta_t, error_leapfrog, 'go', label=\"Leap-Frog, $n=2$\")\n",
    "\n",
    "axes.loglog(delta_t, order_C(delta_t[2], error_euler[2], 1.0) * delta_t**1.0, '--b')\n",
    "axes.loglog(delta_t, order_C(delta_t[2], error_RK2[2], 2.0) * delta_t**2.0, '--r')\n",
    "axes.loglog(delta_t, order_C(delta_t[2], error_leapfrog[2], 2.0) * delta_t**2.0, '--r')\n",
    "\n",
    "axes.grid()\n",
    "axes.legend(loc=2, fontsize=14)\n",
    "axes.set_title(\"Comparison of Errors\", fontsize=18)\n",
    "axes.set_xlabel(\"$\\Delta t$\",fontsize=16)\n",
    "axes.set_ylabel(\"$|U(t_f) - u(t_f)|$\", fontsize=16)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Look at the errors for Leap-Frog\n",
    "\n",
    "They're actually quite large...If you make a quick plot of u_leapfrog vs $t$ you'll see what's happening (and is a good example of an issue we will need to address in future lectures)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5tqRFSyLq5PNBXGMGj8F19OmESdKMOvmS7ZCwDAxvp6egrUG5UkWTCJdQOtGqG3HygiDMKv7k0V38/kPVjn3fyXEADg5GWwEfGXIdfNjJa605Meo6+LCTdxzNmWyR1QvagMrkq9aa4VyJVQtckY/HNZ3pBG0pKyLy44UyrV51zURkxasTTLy6j32Rd518ytuYO1xC6Qt/eNFTKbzoKbSy1Q5n8qZk8oIgzFC2HDzDvlPjVeM/2T7Az3YOoHVFvGxHs3/QzdzD2Xu2UA4mUMMiP15yxbM1aXJqvBB81mi+RNnRXLS4Hajk8vmS22FyWVcLhqpU12itGS+UaU9ZtKcsxiJ18mXXySetqu3//IlX/7PBz+QrTr4y8aqDCCdhqsDhl22HhKlQyi2h9J18uEGZVNcIgjDtnBjNVzUDcxzNff+4lT9+ZGdkfCRX4vCZCcbyZfpHKjHL0aGJ4DMOhETez+OVguOh1w/l3ddesayTkq2DWMbP49f1uCJ/xnPyfkuD7tYknS2JIK6ZKNo4Gi+uiTr5bNEOnHzJ1kHM4k+8+lU0fv+aQtkmlTBI1SihTJhufpS0jMimIZaX00dKKINWw7LHqyAI55H+kVzEfYMbadz6pxn+7ul9kfF9p8YZmijxypHhyHt2Hh8NjncPjEVe73NwMCTyZ1z3fklPR8TJDxXcz7xy+TygUmHjl09e3NPhvi6otHFFvbs1wbyWRGX1qzfR2pFO0J6uiHzJdp1/W9IKFjf5ubxbJ1+JZarimlgJZdF2SHpjlmFEWg374m+FSij9501DqmsEQZhiyrbDM3tOVYn57oExbv7KxqrOjs/3DZIt2mx842RkfPPBM4A72XnodCVj33G8sjr1jZDI93l5/LrF7RwMvd7fqu+GtQs4NVYIyhWH8u71rV/hiryfy/vlkxcu8py8J+6+k5/XkqQznQiqa/zVrh1pi7aUFbQ18AW9LWXRkoxm77mSTdoKlVBGqmvMmouhkp6YJyxFeI/XhJ/Vh0oo7VAJpfSuEQThTdFaR8oIfXYcH+G/P7wjEBafb20+zL/9+mYe33kiMv79bUdxNPw4JvJPeeL+2tHhSMXK1oNDQU36K0eGQ+cdpaczxdJ5afacCDn5k1kWtie5emVX1MkPTdCaNLliWSdQWQDli/wVy3wn7477Tr6nM01n2gomXke8OKerNUFnixU0JfN/tntxTUXk3Z9tSTPi5LXWbiafNElbsTr5sl13xWslkzcivWuswMmHWw1XSiibwskrpe5QSu1WSvUppT7XiHMMjOT58r/s5Ed9Rf7h+QN8/+WjPLZjgOf7Btl2eIi9J8boH8m5kzZ2Yyc5BOFciDtpn76T45EeLD4PbTnCQ1uOVI1/+sFXuO3Pno70Z9Fa84UfbOcfXjjIv7zeH4w7juYbzx8E4Duhz3IczY9fccX9mb2DgRBqrXlq90mWzUvjaNi0/0zwns0HznDrpYtpSZgxkR/hymXzuGRJR9TJnxpn7aJ2Vi9s4+RYIYhNjpzJsbK7leVeffsxL7IZKmgWtidZ1uWO+07+1HgRpdxYZn5bMpi09Ve7dsXiGv9eOj2Rz5ccyrYTrHxtTVm0JNyeMxPFMoWyg9Z4i6FqTLwmjECc/Tr5kl0R+aRpBKWTJafi5N24plYXSlfk6/17mAqshn0yoJQygb8BbgeOAluUUg9rrXe++TvfGidG83xz02EmijY/6Dv7RydNg3TCoCVperPrJi0Jk1TC8J5zH6eTJinLIGEawdJkf5FE0jKDBkOWqUhZlfcbSnkLHSBpur/eJUyFoRRKgaFUsEIuaRkowP8rtgxFyjJQSk3lf6I5TaFskzDcOuYwI7kSpqGCPuA+J0bz5Io2qxe2RcYPDmY5MJjllnULscyKP9rVP8pzewe565plLO5IB+MvHx7i7zL7+PWbVnPTRQuD8Rf2DfKZB1/ho9cs53MfuDT4u35q90nu+8etrJzfyj//9k10tyXdzzlR5q+2vYZSsGJ+Czdd6H7WT17vD+KV/7Oxjy988DIAHttxglePDJNOGHz1qX386/XLMAzF03tOcWAwy6VLOnhq90kGRvIsmZdm88EzHB/J86vXX8C3Nh0ms/skd65fxq7+MU6MFvjSR67kS/+yk+f7Brn98h6OD+c4Npzj3797DcO5Ets8kc8VbfpOjnPHFUso2A4v9J32Jh8VfSfH+dD6pUHp46HTE1y+rJOjQxOsnN/Cck/Mj3u18sN5zZJ5adq8/zdPhTL57tYklmnQ3ZYMnLw/MdtVN65JBA3EsgU74uR9Ic6FFlfVrpN3gjw+aRlBjFO0Hdq99sNuCWU4kw/HNZUSSqUqTt4dm6UiD1wH9Gmt9wMopR4E7gKmVOSvWtnFzj+6gyc2PsW119/MSK5EtlB2/xTLjBdssoUy4/kyuZLt/im6f9ETRZuJok2hbFMoOYzmyhTK/mscCiWbkuNQtnXDs7MwSdMA5bopR7v/IFKmPxmkcLQmXyiSfOZnpL0vGFOp4Fq1hpbQl1Sx7AR9OMJfYIWye49F2wnarrYk3O57E0U7qDTwqxMKZYexfIlswa00mNfiOqeyrRnOFRmeKGGZigVtKRa0J9HabSg1OF5Ea83izjQ9HSmSlsHASJ7+kTz5ks3y7hZWdLfQkU5wdGiCQ6cnGMoWWTm/lbWL2lnSmebQmSwv9+UYeeZnXLCgjcuXdnLhojYOnZ7gtaPDvDEwxoruFjasms/6lfM4NpTjxf2nee3oCPNaEtyybiG3rFvE0ESRx3eeYOuhIUxD8Z6LF3Hn+qUopfjeS0d5bu8pHA3XrZ7Px29cxbKuNF979gA/3TGA1rBmYRu/e9tF3Lh2IX/xxB4e2noER8NfbdzL5z5wKXdfu4K/3tjHVzN9ADyx6wS/976L+cR7L+KhrUf4Lz/cTnva4v8+s58z2SJ/8ovvYNOBM/zW/3uJVQvaOHx6gv/wj1v5p9+8nqNDOe5/rcA7ls8jWyjzn77zKj/59C0YSvFff7SDK5Z1cumSTr7+3AF+acNK1ixs409/tpsLF7XxO70X8dnvvsoTu07wC1cs4evPH6CnM8Vf/+q1vO/Pn+Z7Lx3hk7eu40evHKc1afL5D1zKY9sH+On2Ae5cv4yndrtRzS9c3sNjOwZ4Yd8g4JZOArxr9XxOjOb5xvMHKZRt3hgYxdFwhXetRdvh4GCW7rYkI7kSFy5qD+rYD57OctnSDg6fmeCGtQtYMs/9cjwecvKXLEqjlGJxZyo08VpkgfflN781GSyYGs4VSVqueQtX11QmXq1ge77xYrni5JNWMDk6UbQD1167Tt4OKmuSlhFpX1Cpk69Uy5S9LzhwnbyfIpQcHYyb3rkbqS2NFvnlQPh3zKPA9eEXKKXuA+4D6OnpIZPJvO2T5SeyvLblhciYAXR6fwAwvT9pzoLh/an8J3K0pmRD0XH/ohwNjoay4/4llRz32NGuM7cdTVkTjPu/kjka7NB4GNv7HL/5nW8+He2es+Q4oMEwwC5prISm5JQp2v6XAVieMyw5JQo22Da0G2BZ7ucVHSjkYMSb/W81oNOEQglGcppTNiRMSJuKThNKBTiZ1RwuaxKGoiUBbaailIdjY5o9JY1pQHtC0ZZQODYcHh9n+2H3fuelFJ1J97eYIwNZXjvkfmF2pw3mpxWtJhw7mePVQw4TJVjYoljUarCqDU4NDbP96BmyJehOKXpaHFbNNziZHeXhbcNkS5A2YfU8g1uWGZzK5Xnk1SN8Z+sRTAVr5hn8wiqLobzNkzuO80MvlljZYfDB1RZ5W7Nl38kgo16QVty5NkHagqcOD/Gpb7uC1mrBh9YkWNlh8Mj+HL/3HXexj6ng9gssrltq8d09Rb74g+186cfbyZXh3cst7l6X4KHdRf7s8T18+4W9HM9qrlxo8h+vSvDYQfjuS0fZfuA4+4YdFrcqPnOlw47TCf721SE+9tePc3zcwVKaX7+oyFhR88cv5vnN+zfSYinOZMt8ar2iO3WGRw3NZ/7xOa5fYtJ3ssgnrk7RPbKXRS2Krzy8jcEDKZ7dm+MX1yU4unMrl803eODZvVysj/LwtgmuWmjy0ovPc2W3wxM7+vnZk0/xo615VnUa7Hz5RZaoIs+eKPHDxzby8L4SaRNO7nkZa9Q1CN98JMPBUfcf7djhnWRL7t/99zduoivliWj/Po7kDwKwccvrlPvfcA3WmeP8/LlTdCYVW3ftJ2Me40zOxh4/QyaTIWnn2XtkgEwmw/7jOUwFmUyGwliB/jM2mUyGXfsKtJqap59+mtP9RfIlh8c3PsUrh1yRf2XLixwadMX6qWd/zmDOvdY3tr8S/D+2ZdurHG93xfpA3x62jLoVRTt37+Wp4kEKJYf+Y0fIZE6AXebgkWNkMoMMj03QyQSZTIbsWI7smHt9/SfyFPKaTCbDyRMFJvLutR44WEThjh864H4ZZZ5+hnI+Oyn9q0ejRb5W5hD5ytJa3w/cD7Bhwwbd29v7tk+WyWSYzPtnG3Ppfl0XZUbuWWt39WN3azISxTiO5sjQBAvbU8Gv6P74GwNjdKStYPm7P7710BCO1ly3en7wWY6jeWbvKU6M5vnQ+mVBrPOfHc1Pdwyw7fAQv3r9KtZ4sc5vaM33Xz7GNzcd4rfecyHvv2IJAB95v+Zbmw/zx4/s5NdvWsV/+dBlWKbBB4GrnjvAHz2ykwsXtfGd37qRhe0p7gS6l+/jfzz6Bqah+Ow7W7j7A7cCUOx2xwH+Y++F3HvHpQAMtrmfc2DUrUj57C/fjFKK32s7zBd+8DrfOpAgaRX44i+/hwXtKUa6jvHpB1/h+fFFZEuHuO+Oa+m9ZDEsPcnT39jCePc6+oZf5ZPvvYje3kuYf9Ew393zPKrnEo7v2se71qa47dbruWw4x9+8shFj0VpKzhhdrQPcfcd7KdoOf/TiY5jzV9LelQa289H33cSK7lYWbXoCo3MRF1y2CjY+z63Xraf38h7WbH8O3ZLghps3kP3pT7n20rX09q7joWMvsXtgjN7eXv5oa4bLlnXS23stL0zsYuvJg/T29vLgkZdYXBqnt/c9HEkd5J/37uDqd93Ea+WDqD19vP/WXlr2nuKrr2zhsvXXuNn/y9u45cbrMQ0FL2RYe/GlXNLTCc8+y7Xrr+T2y3vgiUdZdsFqbr7lIvRjP+GSi9bS23sRHZs3smDRfHp7r8batJEVS93jv9+3iWyhTG/vzXxj/2b0RJHe3nfzxPDrbB8aoLe3l2fGdpLqP0Jvby995n7YvYsbb3432zY935D/nxst8keBlaHHK4DjdV4rCHXxs9AwSikWtKeqxg1DsWpBW83xy5d11hy/bs38muO9lyyuOf7Bdyzlg+9YWnU9d79zBXe/c0XV+MeuX8Uvb1gZyfIBfuPda9iwupsL5rfS1ZoMxv/DLWsxlKKnM03H0J5g/DffvZZN+90M/XdvWxeMf/zGVXx782H2nhznD++o5Px3v3M5f/XkXrYfG+WXNqwI/nu9/4oldLUmeODnh5jfluTd3pzBTRcupCNl8Sc/eQNHQ++l7v1fsWwenWmLR1/vZ/eJseDel85Ls7gjxStHhtl3KssVyzpRyp2jWrOwjd0nxoIdmJbNc3P31QtaOTg4EZRPrpzvji/ramHPiTFOeu0MerwIZ1F7iufG3KhocLzAQi+ucbfnc8gVbYZzRbpa3PHOFrd3/Gi+xFjeXe1qhOZesoUyE96EbFvKRFGJa/yVr+mEO4eTNA3yJSdYEJUKTbBGetfUaGtQDk+8hurny44TxDTnI5NvdHXNFmCdUmqNUioJ3AM83OBzCsKMJC7wPutXdEUEHtwvht+8ZS3/+qplkXHDUHzt3g38+JM3B5kxuOLy1Y9dy5c/eiU3hyZ5U5bJb79nLYaCf3fzmmA8nTD56DXLAbhz/dJAjJKWwW2XLebUWIHu1gRXregC3DmhGy9cwGM7TqC1m8f713n1yi62Hhpi98AYV3olj+AucNo9MOZV1rQFvyGtXtDGwdPZYLXrym73t6plXS0cH84HOfsSrwf84s40o/kyY/kSo/kyC70vqvltrpifmXDnguZ5G4P4Ij+Sc0W+09uuz58cHS+UyRYrmbxfJx+feAVIJQzyJTuYZK1k8ma0d02orUGlQVmlhDIRK6H0V8Ka3n/3cgNbGzRU5LXWZeCTwGPALuAhrfWORp5TEJodpVTNL4x1PR187PpVVeP33rSaZ//wVi5bGv0t5uM3rOKC+a38ynUXRMbvuNKNmd5z8SI3yvDwvzwSpivsPlet7OLoUI6i7XDF8pDIL3EnVnceHwl6zQBBGeWegTHmtyWDSG1ZVwu5kjuBCwSTsYs8UfdX0Pq/jXR7X4xD2SIjuRJdnrj7oj6aKzGWLwUbb7clKyLvO/nWWJ2833LYF/60V4RQCJy8J/6hiddo75rQeKyEsrKRd2hCtgmqa9BaPwo82ujzCIJQG6VUUKIYZu2idp75g/dWjb/n4sXcsHY+98TE3y/dvHL5vEAEAa4JCf4VoTjskiVu64HB8WKwQhUIyiif6zvNyu7KdS3vckX95UNDAMFuTos6XFHf1e+K/4J2r7rGi23OZF0n3+U5+Xktrqz5Tt6PacJxzXix7JU2Vxx4OK7xnXw64cY1vmv3Wx24JZSVfvLBb0JmtHdNuISyZLv18GWn4vD9L9FG9pRvuMgLgjC7aEmaPHjfjVXjFy5q4+qVXXwoNhfxjhXzUApaEyZrQnMhl3oiD0ScvF9GOThe4Pq1lbkQf+HTy4eHSZruAiYIibzn5Bd6Iu+vIxgYyZMr2UHkVcnky4wXysHr/d8YxvNlr5d85YuqJWGSL1XiGj8KS1lmzbgmZRmMF8porb3eNaFMvlzZHCRcQgmuY7dDJZRN4eQFQWgOlFL88BM3V413pBNc0tNBZ0siUuW0sruVloRJrmRHnXxokZmfx0NF5A+fmaCnVQWTx4vjTr7Ny+Q9Ufe7Ws6rE9f41U/+4sPxYtnbFaoif61Ji4liuZLJJ8NOPhTXeOLvT7z6k6zR3jXRzUSAwLmXHU3Z1oGDD5y8iLwgCDOZv/nYtYEr9TEMxcU97bx+bITVCyti3p6yWNSR4tRYIaisAVjQlnQXGZUdutOVz1rQnsJQ8Ea/n8lXHLuh4MApV+T9jD6dcFeZj3pxjZ/J++f2nXx4pbO/z2tVXONNsAZxje/kE+5CQj9/r5nJ2+66Eqg49rKjq6puQJy8IAgznLBTD3PjhW4LiHgJ7OoFra7Ih5y8P3dwYDBLd6oi8qahmN+WYnC8QNIyAnE2DUVXa5L9g26HSz+TB9fVByWUMZH3V8K3pkJxjbfPa67oCnQ6UZl49XvaQGXiNXDy3niyRiZftp0gpvHFvGw7dZx846prROQFQWgYf3jHJTX7MK1e0MaWg0ORRWkAy7rSrsino9VDiztSQY18+PO6WxNB62I/rgE3zz81VqRoO0F8A3jtht2WHW3J2k4+6W3mAW5ccybrBJOskbYGISefCGfyoe3/whO74Lp7d+LVF//ZXycvCMIcpl6jvasv6GJ+W7Kq6sdfMBV28lCZfI0vfpvflgwWJYWdfGdLIuhoGY5rOlIW4wW3t1VreOI1aTHhTbz63SfBzeDzZTtw8v5zKcuk6G08AkT2ePUnV0t2ZccoK1QPX3acpupdIwiCUMWvvOsCfvGaFUGW7bPUE/1wJg+VyVc/j/fpDi0iCy8om9eSoO+kW4oZFvm2lMngeNF18uFMPmEyMJIjV7Qj5aFpy6QQLqGMdKG0qzN5y3fsjlddE3XsZTs68SpOXhCEpsQwVERMffxa+bjI+05+YQ0nD65YhksiO9OJSgfKVDSuyRbKTBSjTj4c17SEVhJXVdfEulDGnbz/s2S7UY4v+onQuO1UtgWUOnlBEOYUt1++hP2nsqxKD0TGF9Vz8p7Id7UmItFQZ0tosjUc16Qtxgpuq+Gwk097E69uXBMW+XidfGXFa8nWgcMPV9eAl73bDgmjuoSy5GhajfNXXSNOXhCEGcP8tiSf/+BlVeWY/mYsC9tiTt6LaMKTrvHHkbgmaTGWL5Er2VEnnwg5+WTMyZdrr3gFgh2uwg3KwO1B72iqSiXLtsZ2nKC08nxU14jIC4Iw4zm7k4+Ohytq4tU1/sYg8eoafzOhtBXN5G1HB2Iej2X8SCjcoAwqG4VbZrxO3pFMXhAEIc47ls/j7mtXBC2RffxOlF0xJ99Zx8mHj6N18u740EQx5uTd4+Fc0d3a0xNlP5sfjzl5X+z9RmeV6pp4CeX5W/EqIi8IwoynJWnyZ790FYs7o1u6+dU181rrxzXh7D1yHHPy4DY7i0+8AozkyoGwQyWbH/e2GUzFMvlssXaMU/YmXq1YVu+IyAuCIFTjV9f4G4b4+BFNS6KyUTdERT5aJ+879lJk4tUX89FcKcjjoeLY404+vCk4ULXoqezVz8cblImTFwRBqIGfyfuxjY9fXROOZ8BdDOUT710DoDW0JMOLoXwnX4q0ZqiIvB15bMUy+WRsMZRfQlmJa6R3jSAIQl060wn+8p6ruXHtgsi4H9fERT7i5Ou5+kR1Jj+aK8XiGn/i1Y1rfJH3J2QnvLjGj2X8bL5sa0q2DsT9fDh5EXlBEGY1d129vGrMj2s60lGH3xaabI32k7dCx9UiP5IrBRU+UCuuiS568rtZVqpr/LYGbgmlFSuhtKWEUhAE4dzxHXx1XFMR/XpOPhUWeSsU14TGfcful1amTPc5X+yzfowTa1Dm9q7RNUorJZMXBEE4ZyzTbUlcHdfUdvJni2vKjo7ENckgrvGcfKx9Qc6Pa7zHZqx3TbWTF5EXBEF4S6xZ2BbsJ+sTra4JRTQ1Km2AqhYHPkEJZXyRlOVn8tG4Jt67prrPvGTygiAIb4nv/vaNVe0RUpYRRCfhDphh916rTt5/r084kzdUxbFX6uSjcU20d011q2GprhEEQXiLhJ23j1KKtpSFjmlq2NXXd+/Vgj+eL0fq8P0vkCCuMaITr8Wyg9ZUtTWQ6hpBEIQpoj1lVa0wTScMlPLr5KO9a3xSVrXgjxWiK2ErJZReW4NYTxt/o/BELKu3Hadh4blk8oIgzCnaU1aksgZch+/HNOG4JhXZJao6rimWnUDYoSLegcgH7Qv87pRuqWSwx6uS6hpBEIQppS1lRSprfPwKm3gO77epr5XJx48TwcSrX10TjWXy3uYj/mPDUBhKMnlBEIQp46PXLKdsVy8+8mOasJNXSpGyDPIlJ9rWoIZ7d4+jbQ0Ssd41flwTnhC2DEMyeUEQhKni125YVXO81Vv1Gp+wTVmmJ/IVMbdMA9NQ2I6OOnkjFtfEWgr7Im+Gvhj8z2kUEtcIgiAQcvKxKMePb8KZPFRvIAJu/GIZqsrJK6VImCrI5BMRJ68aWicvIi8IgkAlk29JxEXe39c1Oh7sBmVFZTRhGlWZPLixTODkQyJvmmrm9q5RSv0vpdQbSqnXlFI/UEp1hZ77vFKqTym1Wyn1/klfqSAIQgOpTLzGRD60eXcY/3HKjIu8ChqU+fENuIKfi5VQgufkZ3Bc8zhwpdZ6PbAH+DyAUupy4B7gCuAO4KtKqerpbEEQhBlCS9IiaRkRlw2huCYm8hUnr6rG/cVWYZefMOs4+ZmcyWutf6a1LnsPXwRWeMd3AQ9qrQta6wNAH3DdZM4lCILQSFoTZtB1MozffTKVqB3XJKucfNSlh4/9TH62Vtf8BvAd73g5ruj7HPXGqlBK3QfcB9DT00Mmk3nbFzA+Pj6p98825tr9gtzzXGE67nm1srljlao678RYHoC9b+wkM7QnGC/lcwCMDp+JvKdcLATHLzz3DIZXaG+XipwedjcZeWPXDlpO7wagWMhzrL+f8ZZyQ+75rCKvlHoCWFLjqS9qrX/kveaLQBn4pv+2Gq+v+VWltb4fuB9gw4YNure39+xXXYdMJsNk3j/bmGv3C3LPc4XpuOd6Z/vW4a1sHzzBO69eT+8li4Px+Tue58jYMEt7FtPbe20w3vlShlO5LIaCW9/73mC8bfNTbkwzluXq9evpvdT9rI6tGRYumkd7+0hD7vmsIq+1ft+bPa+Uuhe4E7hN66Dtz1FgZehlK4Djb/ciBUEQpot61TWpWHthn2ABVCzGsUzFRCHaghj8TH7mVtfcAfwh8GGt9UToqYeBe5RSKaXUGmAdsHky5xIEQZgO6tbJ18nk62b1hhG0NYhPvM7kfvJ/DaSAx5WbO72otf5trfUOpdRDwE7cGOcTWmt7kucSBEE47/hOPh138tbZnHw0tbZMRc7fTCRWWjlje9dorS96k+e+DHx5Mp8vCIIw3QRxzTk6+URs824fyzQolL3qmkhc09jqGlnxKgiC8Cb4ZZX16+RrO/lkzMnHWxmEj2dsnbwgCEKzk6o38Vovk3+TidfgOOTyTUNRnqkTr4IgCM3O8q4W2lMWHeloup2sk8kHPeTjTt40ql4DjXfy0mpYEAThTfjwVct47yWLq3raJE3T+1kvromOxytqwse5ksQ1giAI04JhKOa1JqrG/YnYRMyxJ+tV14QimkjjMsnkBUEQZh5BP/lYVu87+USdqhtw2wsHx4Yh/eQFQRBmGvUyeb8rZaJGCWXwGqmuEQRBmNn41TXxuKbeYqhEvUzelOoaQRCEGUeqXv18nbimXgmlOHlBEIQZSLAYqk51TdzhW3VKKM0ZvjOUIAjCnKRuJl9v4rVOXCNOXhAEYQbir4CtqpO3/MVQ8Tr56FaA4XFx8oIgCDOMlnrbAvpO3ohPyFYeh58SJy8IgjADuemiBfzxR65k/fJ5kfF6cU3Q7sBQeK3ZAb+ffOOqa6StgSAIwtsgZZl8/IZVVeN1+8kb9cbFyQuCIMwa/Fim3orXeJ95t05eRF4QBGFWkKyzSMqq6/DFyQuCIMwa6m7kbajITx9TuU5e68YIvYi8IAjCFFK/QZn72IyLvBffNMrLi8gLgiBMIUEmXyXm9fZ+dccb1YhSRF4QBGEKqbf9X6LOjlG++DeqR5mIvCAIwhSSqDfx6pdQGtUTryBOXhAEYVZw9sVQtbcFbFSBjYi8IAjCFFK/Tr72xKs4eUEQhFlE/T1effGvXV3jSAmlIAjCzKerNUnCVCzuSEXGp8vJS+8aQRCEKWRRR4oXPncbC9uTkfEgk69qQdzYTF5EXhAEYYpZFHPxEK6Tj7c78Jy8lFAKgiDMXuqvePWcfIPOOyUir5T6rFJKK6UWhsY+r5TqU0rtVkq9fyrOIwiCMFupTLzW7mkzY+MapdRK4HbgcGjscuAe4ApgGfCEUupirbU92fMJgiDMRs7Wu6ZRnSinwsn/b+APiPbXuQt4UGtd0FofAPqA66bgXIIgCLMSy6xdQjmjnbxS6sPAMa31q+HtrIDlwIuhx0e9sVqfcR9wH0BPTw+ZTOZtX8/4+Pik3j/bmGv3C3LPc4VmvOfj427qfvrUqci97RgsAzA+kWvIPZ9V5JVSTwBLajz1ReALwC/UeluNsZrfU1rr+4H7ATZs2KB7e3vPdkl1yWQyTOb9s425dr8g9zxXaMZ7PnQ6C89lWLZ0Cb29Vwfjyb5B2LqJVLqlIfd8VpHXWr+v1rhS6h3AGsB38SuAl5VS1+E695Whl68Ajk/6agVBEGYpwc5QdaprZlxbA63161rrxVrr1Vrr1bjCfq3WegB4GLhHKZVSSq0B1gGbp+SKBUEQZiHBzlBV2wL6mXxjVL4hi6G01juUUg8BO4Ey8AmprBEEYS5j1dk0JKiumYkTr2E8Nx9+/GXgy1P1+YIgCLMZ6yy9a6TVsCAIwiwm2BmqXiYvbQ0EQRBmL8HOUOd5xauIvCAIwnngrE5+pmfygiAIQn2UUvz+7Rdz62WLI+NWgzcNEZEXBEE4T3zqtnVVY6Y5Q+vkBUEQhMkjmbwgCEITM2NXvAqCIAiTJ3DyUkIpCILQfIiTFwRBaGIaXV0jIi8IgjCNiJMXBEFoYqS6RhAEoYkxDIVS4uQFQRCaFstQ4uQFQRCaFdNQ4uQFQRCaFcswcBpk5UXkBUEQphlx8oIgCE2MZPKCIAhNjDh5QRCEJkacvCAIQhNjmuLkBUEQmhbLMKR3jSAIQrPynosXsWae2ZDPFpEXBEGYZv77h6/g/asTDflsEXlBEIQmRkReEAShiRGRFwRBaGJE5AVBEJoYEXlBEIQmZtIir5T6lFJqt1Jqh1Lqf4bGP6+U6vOee/9kzyMIgiC8dazJvFkp9V7gLmC91rqglFrsjV8O3ANcASwDnlBKXay1tid7wYIgCMK5M1kn/zvAV7TWBQCt9Ulv/C7gQa11QWt9AOgDrpvkuQRBEIS3yKScPHAxcItS6stAHvis1noLsBx4MfS6o95YFUqp+4D7vIfjSqndk7iehcDgJN4/25hr9wtyz3MFuee3xqp6T5xV5JVSTwBLajz1Re/93cANwLuAh5RSawFV4/U1GzNore8H7j/bdZwLSqmtWusNU/FZs4G5dr8g9zxXkHueOs4q8lrr99V7Tin1O8D3tdYa2KyUcnC/jY4CK0MvXQEcn+S1CoIgCG+RyWbyPwRuBVBKXQwkcX/deBi4RymVUkqtAdYBmyd5LkEQBOEtMtlM/uvA15VS24EicK/n6ncopR4CdgJl4BPnqbJmSmKfWcRcu1+Qe54ryD1PEUo3qIexIAiCMP3IildBEIQmRkReEAShiWkKkVdK3eG1T+hTSn1uuq+n0SilViqlnlJK7fLaSXx6uq/pfKGUMpVS25RSj0z3tZwPlFJdSqnvKaXe8P6+b5zua2okSqnf8/5Nb1dKfVsplZ7ua2oESqmvK6VOevOZ/th8pdTjSqm93s/uqTjXrBd5pZQJ/A3wAeBy4Fe8tgrNTBn4fa31ZbhrFD4xB+7Z59PArum+iPPIXwI/1VpfClxFE9+7Umo58LvABq31lYCJ2x6lGfkH4I7Y2OeAJ7XW64AnvceTZtaLPG67hD6t9X6tdRF4ELetQtOite7XWr/sHY/h/o9fc0VxM6GUWgF8CPjadF/L+UAp1Qn8K+DvAbTWRa318LReVOOxgBallAW00qTra7TWzwBnYsN3AQ94xw8AH5mKczWDyC8HjoQe122h0IwopVYD1wCbpvlSzgd/AfwB4EzzdZwv1gKngG94EdXXlFJt031RjUJrfQz4U+Aw0A+MaK1/Nr1XdV7p0Vr3g2vkgMVT8aHNIPLn3EKh2VBKtQP/DHxGaz063dfTSJRSdwIntdYvTfe1nEcs4Frgb7XW1wBZpuhX+JmIl0HfBazB7V7bppT6tem9qtlPM4j8nGyhoJRK4Ar8N7XW35/u6zkP3Ax8WCl1EDeSu1Up9U/Te0kN5yhwVGvt/5b2PVzRb1beBxzQWp/SWpeA7wM3TfM1nU9OKKWWAng/T57l9edEM4j8FmCdUmqNUiqJO1Hz8DRfU0NRSincnHaX1vrPp/t6zgda689rrVdorVfj/h1v1Fo3tcvTWg8AR5RSl3hDt+GuIm9WDgM3KKVavX/jt9HEE801eBi41zu+F/jRVHzoZNsaTDta67JS6pPAY7iz8V/XWu+Y5stqNDcDHwdeV0q94o19QWv96PRdktAgPgV80zMw+4F/N83X0zC01puUUt8DXsatINtGk7Y3UEp9G+gFFiqljgL/DfgKbifff4/7hfdvpuRc0tZAEASheWmGuEYQBEGog4i8IAhCEyMiLwiC0MSIyAuCIDQxIvKCIAhNjIi8IAhCEyMiLwiC0MT8f06bE69sctVeAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N= 100\n",
    "t_leapfrog, u_leapfrog = leap_frog(f, t_span, u_0, N, start=euler)\n",
    "## Your plotting code here\n",
    "plt.figure()\n",
    "plt.plot(t_leapfrog, u_leapfrog)\n",
    "plt.grid()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### General Linear Multi-Step Methods\n",
    "\n",
    "Leap-frog is perhaps the simplest of multi-step methods but all linear multi-step methods can be written as the linear combination of past, present and future solutions:\n",
    "$$\n",
    "    \\sum^r_{j=0} \\alpha_j U_{n+j} = \\Delta t \\sum^r_{j=0} \\beta_j f(U_{n+j}, t_{n+j})\n",
    "$$\n",
    "If $\\beta_r = 0$ then the method is explicit (only requires previous time steps).  Note that the coefficients are not unique as we can multiply both sides by a constant.  In practice a normalization of $\\alpha_r = 1$ is used.\n",
    "\n",
    "For example:  our Leap-frog method can be written using $r=2$, $\\alpha = \\begin{bmatrix} -1 & 0 & 1\\\\\n",
    "\\end{bmatrix}$, $\\beta = \\begin{bmatrix} 0 & 2 & 0 \\\\ \\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Example: Adams Methods\n",
    "\n",
    "$$\n",
    "    U_{n+r} = U_{n+r-1} + \\Delta t \\sum^r_{j=0} \\beta_j f(U_{n+j}).\n",
    "$$\n",
    "All these methods have $\\alpha_r = 1$, $\\alpha_{r-1} = -1$ and $\\alpha_j=0$ for $j < r - 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Adams-Bashforth Methods\n",
    "The **Adams-Bashforth** methods are explicit solvers that maximize the order of accuracy given a number of steps $r$.  This is accomplished by looking at the Taylor series and picking the coefficients $\\beta_j$ to eliminate as many terms in the Taylor series as possible.\n",
    "$$\\begin{aligned}\n",
    "    \\text{1-step:} & ~ & U_{n+1} &= U_n +\\Delta t f(U_n) \\\\\n",
    "    \\text{2-step:} & ~ & U_{n+2} &= U_{n+1} + \\frac{\\Delta t}{2} (-f(U_n) + 3 f(U_{n+1})) \\\\\n",
    "    \\text{3-step:} & ~ & U_{n+3} &= U_{n+2} + \\frac{\\Delta t}{12} (5 f(U_n) - 16 f(U_{n+1}) + 23 f(U_{n+2})) \\\\\n",
    "    \\text{4-step:} & ~ & U_{n+4} &= U_{n+3} + \\frac{\\Delta t}{24} (-9 f(U_n) + 37 f(U_{n+1}) -59 f(U_{n+2}) + 55 f(U_{n+3}))\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def AB2(f, t_span, u0, N, start=RK2):\n",
    "    \"\"\" calculate fixed step Adams-Bashforth 2-step method with a single step starter\n",
    "        reuses previous function evaluations\n",
    "    \"\"\"\n",
    "    t = numpy.linspace(t_span[0], t_span[1], N)\n",
    "    delta_t = t[1] - t[0]\n",
    "    u = numpy.zeros(t.shape)\n",
    "    \n",
    "    u[0] = u0\n",
    "    # use a single-step multi-stage method to start\n",
    "    t_start, u_start = start(f, (t[0],t[1]), u0, 2)\n",
    "    u[1] = u_start[-1]\n",
    "    \n",
    "    # set initial function evaluations\n",
    "    fn = f(t[0], u[0])\n",
    "    fnp = f(t[1], u[1])\n",
    "    for (n, t_np) in enumerate(t[2:]):\n",
    "        u[n+2] = u[n + 1] + delta_t / 2.0 * (-fn + 3.0 * fnp)\n",
    "        fn = fnp\n",
    "        fnp = f(t_np, u[n+2])\n",
    "    return t, u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Use 2-step Adams-Bashforth to compute solution\n",
    "f = lambda t, u: -u\n",
    "t_span = (0., 10.)\n",
    "u0 = 1.0\n",
    "\n",
    "N = 20\n",
    "t, u_ab2 = AB2(f, t_span, u0, N, start=RK2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t_exact = numpy.linspace(t_span[0], t_span[1], 100)\n",
    "u_exact = numpy.exp(-t_exact)\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "axes.plot(t_exact, u_exact, 'k', label=\"True\")\n",
    "axes.plot(t, u_ab2, 'ro', label=\"2-step A-B\")\n",
    "\n",
    "axes.set_title(\"Adams-Bashforth Method\", fontsize=18)\n",
    "axes.set_xlabel(\"t\", fontsize=16)\n",
    "axes.set_ylabel(\"u(t)\",fontsize=16)\n",
    "axes.legend(loc=1, fontsize=14)\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Adams-Moulton Methods\n",
    "The **Adams-Moulton** methods are the implicit versions of the Adams-Bashforth methods.  Since this gives one additional parameter to use $\\beta_r$ these methods are generally one order of accuracy greater than their counterparts.\n",
    "$$\\begin{aligned}\n",
    "    \\text{1-step:} & ~ & U_{n+1} &= U_n + \\frac{\\Delta t}{2} (f(U_n) + f(U_{n+1})) \\\\\n",
    "    \\text{2-step:} & ~ & U_{n+2} &= U_{n+1} + \\frac{\\Delta t}{12} (-f(U_n) + 8f(U_{n+1}) + 5f(U_{n+2})) \\\\\n",
    "    \\text{3-step:} & ~ & U_{n+3} &= U_{n+2} + \\frac{\\Delta t}{24} (f(U_n) - 5f(U_{n+1}) + 19f(U_{n+2}) + 9f(U_{n+3})) \\\\\n",
    "    \\text{4-step:} & ~ & U_{n+4} &= U_{n+3} + \\frac{\\Delta t}{720}(-19 f(U_n) + 106 f(U_{n+1}) -264 f(U_{n+2}) + 646 f(U_{n+3}) + 251 f(U_{n+4}))\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Use 2-step Adams-Moulton to compute solution\n",
    "# u' = - decay u\n",
    "decay_constant = 1.0\n",
    "f = lambda t, u: -decay_constant * u\n",
    "\n",
    "t_exact = numpy.linspace(0.0, 10.0, 100)\n",
    "u_exact = numpy.exp(-t_exact)\n",
    "\n",
    "N = 20\n",
    "# N = 10\n",
    "# N = 5\n",
    "t = numpy.linspace(0, 10.0, N)\n",
    "delta_t = t[1] - t[0]\n",
    "U = numpy.empty(t.shape)\n",
    "U[0] = 1.0\n",
    "U[1] = U[0] + 0.5 * delta_t * f(t[0], U[0])\n",
    "U[1] = U[0] + delta_t * f(t[0], U[1])    \n",
    "integration_constant = 1.0 / (1.0 + 5.0 * decay_constant * delta_t / 12.0)\n",
    "for n in range(t.shape[0] - 2):\n",
    "    U[n+2] = (U[n+1] + decay_constant * delta_t / 12.0 * (U[n] - 8.0 * U[n+1])) * integration_constant\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "axes.plot(t_exact, u_exact, 'k', label=\"True\")\n",
    "axes.plot(t, U, 'ro', label=\"2-step A-M\")\n",
    "\n",
    "axes.set_title(\"Adams-Moulton Method ($f=-u$)\", fontsize=18)\n",
    "axes.set_xlabel(\"t\", fontsize=16)\n",
    "axes.set_ylabel(\"u(t)\", fontsize=16)\n",
    "axes.legend(loc=1, fontsize=14)\n",
    "axes.grid()\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Truncation Error for Multi-Step Methods\n",
    "\n",
    "We can again find the truncation error in general for linear multi-step methods:\n",
    "$$\\begin{aligned}\n",
    "    T(t, u; \\Delta t) &= \\frac{1}{\\Delta t} \\left [\\sum^r_{j=0} \\alpha_j u_{n+j} - \\Delta t \\sum^r_{j=0} \\beta_j f(u_{n+j}, t_{n+j}) \\right ]\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Using the general expansion and evaluation of the Taylor series about $t_n$ we have\n",
    "$$\\begin{aligned}\n",
    "    u(t_{n+j}) &= u(t_n) + j \\Delta t u'(t_n) + \\frac{1}{2} (j \\Delta t)^2 u''(t_n) + \\mathcal{O}(\\Delta t^3) \\\\\n",
    "    u'(t_{n+j}) &= u'(t_n) + j \\Delta t u''(t_n) + \\frac{1}{2} (j \\Delta t)^2 u'''(t_n) + \\mathcal{O}(\\Delta t^3)\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "collecting terms of order $u^{(p)}$\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    T(t, u; \\Delta t) &= \\frac{1}{\\Delta t}\\left( \\sum^r_{j=0} \\alpha_j\\right) u(t_n) + \\left(\\sum^r_{j=0} (j\\alpha_j - \\beta_j)\\right) u'(t_n) + \\Delta t \\left(\\sum^r_{j=0} \\left (\\frac{1}{2}j^2 \\alpha_j - j \\beta_j \\right) \\right) u''(t_n) \\\\\n",
    "& \\quad \\quad + \\cdots + \\Delta t^{q - 1} \\left (\\sum^r_{j=0} \\left(\\frac{1}{q!} j^q \\alpha_j - \\frac{1}{(q-1)!} j^{q-1} \\beta_j \\right) \\right) u^{(q)}(t_n) + \\cdots\n",
    "\\end{aligned}$$\n",
    "\n",
    "The method is *consistent* if the first two terms of the expansion vanish, i.e. $\\sum^r_{j=0} \\alpha_j = 0$ and $\\sum^r_{j=0} j \\alpha_j = \\sum^r_{j=0} \\beta_j$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compare accuracy between RK-2, AB-2 and AM-2, RK-4\n",
    "f = lambda t, u: -u\n",
    "u_exact = lambda t: numpy.exp(-t)\n",
    "\n",
    "t_f = 10.0\n",
    "t_span = (0.0, t_f)\n",
    "\n",
    "num_steps = [2**n for n in range(4,10)]\n",
    "delta_t = numpy.empty(len(num_steps))\n",
    "error_rk = numpy.empty(len(num_steps))\n",
    "error_rk4 = numpy.empty(len(num_steps))\n",
    "error_ab = numpy.empty(len(num_steps))\n",
    "error_am = numpy.empty(len(num_steps))\n",
    "\n",
    "for (i, N) in enumerate(num_steps):\n",
    "    t = numpy.linspace(0, t_f, N)\n",
    "    delta_t[i] = t[1] - t[0]\n",
    "        \n",
    "    # Compute RK2\n",
    "    tt, U_rk  = RK2(f, t_span, u0, N)\n",
    "    \n",
    "    # Compute RK4\n",
    "    tt, U_rk4  = RK4(f, t_span, u0, N)\n",
    "        \n",
    "    # Compute Adams-Bashforth 2-stage\n",
    "    tt, U_ab  = AB2(f, t_span, u0, N)\n",
    "    \n",
    "    # Compute Adama-Moulton 2-stage\n",
    "    U_am = numpy.empty(t.shape)\n",
    "    U_am[:2] = U_rk[:2]\n",
    "    decay_constant = 1.0\n",
    "    integration_constant = 1.0 / (1.0 + 5.0 * decay_constant * delta_t[i] / 12.0)\n",
    "    for n in range(t.shape[0] - 2):\n",
    "        U_am[n+2] = (U_am[n+1] + decay_constant * delta_t[i] / 12.0 * (U_am[n] - 8.0 * U_am[n+1])) * integration_constant\n",
    "        \n",
    "    # Compute error for each\n",
    "    error_rk[i] = numpy.linalg.norm(delta_t[i] * (U_rk - u_exact(t)), ord=1)\n",
    "    error_rk4[i] = numpy.linalg.norm(delta_t[i] * (U_rk4 - u_exact(t)), ord=1)\n",
    "    error_ab[i] = numpy.linalg.norm(delta_t[i] * (U_ab - u_exact(t)), ord=1)\n",
    "    error_am[i] = numpy.linalg.norm(delta_t[i] * (U_am - u_exact(t)), ord=1)\n",
    "    \n",
    "# Plot error vs. delta_t\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "axes.loglog(delta_t, error_rk, 'ko', label='RK-2')\n",
    "axes.loglog(delta_t, error_ab, 'bo', label='AB-2')\n",
    "axes.loglog(delta_t, error_am, 'go', label=\"AM-2\")\n",
    "axes.loglog(delta_t, error_rk4, 'co', label='RK-4')\n",
    "\n",
    "\n",
    "order_C = lambda delta_x, error, order: numpy.exp(numpy.log(error) - order * numpy.log(delta_x))\n",
    "axes.loglog(delta_t, order_C(delta_t[0], error_ab[0], 1.0) * delta_t**1.0, '--r')\n",
    "axes.loglog(delta_t, order_C(delta_t[1], error_ab[1], 2.0) * delta_t**2.0, '--b')\n",
    "axes.loglog(delta_t, order_C(delta_t[1], error_am[1], 3.0) * delta_t**3.0, '--g')\n",
    "axes.loglog(delta_t, order_C(delta_t[1], error_rk4[1], 4.0) * delta_t**4.0, '--c')\n",
    "\n",
    "\n",
    "axes.legend(loc=4, fontsize=14)\n",
    "axes.set_title(\"Comparison of Errors\",fontsize=18)\n",
    "axes.set_xlabel(\"$\\Delta t$\",fontsize=16)\n",
    "axes.set_ylabel(\"$|U(t) - u(t)|$\", fontsize=16)\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Predictor-Corrector Methods\n",
    "\n",
    "One way to simplify the Adams-Moulton methods so that implicit evaluations are not needed is by estimating the required implicit function evaluations with an explicit method.  These are often called **predictor-corrector** methods as the explicit method provides a *prediction* of what the solution might be and the now explicit *corrector* step works to make that estimate more accurate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Example: One-Step Adams-Bashforth-Moulton\n",
    "\n",
    "Use the One-step Adams-Bashforth method to predict the value of $U_{n+1}$ and then use the Adams-Moulton method to correct that value:\n",
    "$$\\begin{aligned}\n",
    "    \\hat{U}_{n+1} &= U_n + \\Delta t f(U_n) \\\\\n",
    "    U_{n+1} &= U_n + \\frac{1}{2} \\Delta t \\left[f(U_n) + f(\\hat{U}_{n+1}) \\right]\n",
    "\\end{aligned}$$\n",
    "leading to a second order accurate method.  Note this algorithm is identical to \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\__________?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# One-step Adams-Bashforth-Moulton\n",
    "f = lambda t, u: -u\n",
    "\n",
    "t_exact = numpy.linspace(0.0, 10.0, 100)\n",
    "u_exact = numpy.exp(-t_exact)\n",
    "\n",
    "N = 50\n",
    "t = numpy.linspace(0, 10.0, N)\n",
    "delta_t = t[1] - t[0]\n",
    "U = numpy.empty(t.shape)\n",
    "\n",
    "U[0] = 1.0\n",
    "for n in range(t.shape[0] - 1):\n",
    "    U[n+1] = U[n] + delta_t * f(t[n], U[n])\n",
    "    U[n+1] = U[n] + 0.5 * delta_t * (f(t[n], U[n]) + f(t[n+1], U[n+1]))\n",
    "    \n",
    "t_ie, u_ieuler = improved_euler(f, (0.0, 10.), 1., N)\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "axes.plot(t_exact, u_exact, 'k', label=\"True\")\n",
    "axes.plot(t, U, 'ro', label=\"2-step A-B\")\n",
    "axes.plot(t_ie, u_ieuler, 'bx', label=\"Improved Euler\")\n",
    "\n",
    "\n",
    "axes.set_title(\"Adams-Bashforth-Moulton P/C Method\", fontsize=18)\n",
    "axes.set_xlabel(\"t\", fontsize=18)\n",
    "axes.set_ylabel(\"u(t)\", fontsize=18)\n",
    "axes.legend(loc='best', fontsize=14)\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  }
 ],
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