problems_07_odes_part_1.ipynb

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    "### Problem 1: Free Fall with Air Resistance\n",
    "In this problem, you will write a function that uses the four integration methods demonstrated in the lecture demonstration to solve the one-dimensional projectile (free fall) problem with and without air resistance.\n",
    "\n",
    "Consider a baseball thrown straight up in the air. The forces acting on the baseball are gravity, $\\vec{F}_g = - m g \\hat{j}$, and drag from air resistance with turbulent flow $\\vec{F}_{D}=-\\tfrac{1}{2}\\rho v^2 C_d A \\hat{v}$. The radius of the baseball is $r = 0.0366$ m, and $A = \\pi r^2$. The mass is $m = 0.145$ kg, the drag coefficient is $C_d = 0.5$ (appropriate for a batted ball or pitched fastball), and the density of air is $\\rho = 1.16$ kg/m$^3$ (appropriate for PNC Park at an elevation of 250 m).\n",
    "\n",
    "a. Using the forces from gravity and drag in turbulent flow, write down the system of two first-order ODEs for dependent variables $h$ (the height above the ground) and $v$ (the vertical component of the velocity).\n",
    "\n",
    "b. With no air resistance ($C_d = 0$), integrate the equations with the four algorithms used in the lecture example (`Euler`, `Euler_symplectic`, `RK2`, and `RK4`). The integration should stop when the baseball reaches the ground (the same height that it started at).\n",
    " * Calculate the trajectory of the baseball when the ball is given an initial vertical velocity $v_0=50$ m/s.\n",
    " * Plot the results with time on the $x$-axis and height on the $y$-axis (with units!).\n",
    " * Print the maximum height the ball reaches in your simulation, and compare it to the height that the baseball should reach analytically.\n",
    " * What $dt$ do you need to choose for each method to get a result for the maximum height that is within $1$% of the analytical result?\n",
    " * Compare, in one figure, the motion calculated with all four methods.\n",
    "\n",
    "c. Now add air resistance ($C_d = 0.5$), and again integrate the equations with the four algorithms used in the lecture example. The integration should stop when the baseball reaches the ground (the same height that it started at).\n",
    " * Calculate the trajectory of the baseball when the ball is given an initial vertical velocity $v_0=50$ m/s.\n",
    " * Plot the results with time on the $x$-axis and height on the $y$-axis (with units!).\n",
    " * Using the $dt$ values you found for $1$\\% accuracy in (b), plot the motion with all four methods on the same figure.  The maximum height should be less than without air resistance.\n",
    "\n",
    "d. With air resistance ($C_d = 0.5$), increase $v_0$ to $500$ m/s. Use `RK4` and decrease the timestep $dt$ to $0.1$ s or less to accurately deal with the increased speed.\n",
    " * This is an unrealistic speed to throw a baseball, but notice the behaviour on the way down - the baseball reaches its terminal velocity.  What is the magnitude of that velocity?\n",
    " * How does it compare to the commonly reported terminal velocity of a baseball, $33$ m/s?\n",
    "\n",
    "------"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "##########################################################################\n",
    "DISCLAIMER: There is a TON of scratch work in this.\n",
    "##########################################################################"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EULER EXPLICIT INFO:\n",
      "dt: 0.30612244897959184\n",
      "Max height: 16.846468776292078\n"
     ]
    },
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "###########################################################################\n",
    "### FOUR INTEGRATION METHODS: euler_explicit, euler_symplectic, rk2, rk4 ##\n",
    "###########################################################################\n",
    "\n",
    "def euler_explicit_step(func, y, t, dt):\n",
    "    return y + dt * func(y, t)\n",
    "\n",
    "def euler_explicit(func, y0, t, not_prob2=False):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0  # Store initial values\n",
    "    \n",
    "    for i in range(0,len(t)-1):\n",
    "        dt = t[i+1]-t[i]\n",
    "        y[i+1,:] = euler_explicit_step(func, y[i,:], t[i], dt)\n",
    "    print('EULER EXPLICIT INFO:')\n",
    "    print('dt:', t[1]-t[0])\n",
    "    print('Max height:', max(y[:,0]))\n",
    "    if not_prob2:\n",
    "        return y\n",
    "    return y[:,0]\n",
    "\n",
    "###########################################################################\n",
    "\n",
    "def euler_symplectic_step(func, y, t, dt):\n",
    "    return y + dt * func(y, t)\n",
    "\n",
    "def euler_symplectic(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0[0]  # Store initial values\n",
    "        \n",
    "    print(\"y0: \", y0)\n",
    "    for i in range(0,len(t)-1):\n",
    "        dt = t[i+1]-t[i]\n",
    "        # Build up so that, in general, the jth variable is updated \n",
    "        #   with the i+1 position of the (j-1)th variable.\n",
    "        # Requires an extra function call for each variable\n",
    "        # Currently only defined for 2-D.\n",
    "        r1 = euler_symplectic_step(func, y[i  ,:], t[i], dt)\n",
    "        y[i+1,0] = r1[0]\n",
    "        y[i+1,1] = y[i,1]\n",
    "\n",
    "        r2 = euler_symplectic_step(func, y[i+1,:], t[i], dt)\n",
    "        y[i+1,1] = r2[1]\n",
    "    print('EULER SYMPLECTIC INFO')\n",
    "    print('dt:', t[1]-t[0])\n",
    "    print('Max height:', max(y[:,0]))\n",
    "    return y[:,0]\n",
    "\n",
    "###########################################################################\n",
    "\n",
    "\n",
    "def rk2_step(func, y, t, dt):\n",
    "    k1 = dt * func(y, t)\n",
    "    k2 = dt * func(y + k1/2, t + dt/2)\n",
    "    y_new = y + k2\n",
    "    return y_new\n",
    "\n",
    "def rk2(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0\n",
    "    for i in range(0, len(t)-1):\n",
    "        dt = t[i+1] - t[i]\n",
    "        y[i+1,:] = rk2_step(func, y[i,:], t[i], dt)\n",
    "    return y[:,0]\n",
    "\n",
    "###########################################################################\n",
    "\n",
    "def rk4_step(func, y, t, dt):\n",
    "    k1 = dt * func(y, t)\n",
    "    k2 = dt * func(y + k1/2, t + dt/2)\n",
    "    k3 = dt * func(y + k2/2, t + dt/2)\n",
    "    k4 = dt * func(y + k3, t * dt)\n",
    "    y_new = y + (1/6) * (k1 + 2*k2 +2*k3 + k4)\n",
    "    return y_new\n",
    "\n",
    "def rk4(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0\n",
    "    for i in range(0, len(t)-1):\n",
    "        dt = t[i+1] - t[i]\n",
    "        y[i+1,:] = rk4_step(func, y[i,:], t[i], dt)\n",
    "    return y[:,0]\n",
    "\n",
    "###########################################################################\n",
    "####################### TESTING ALL FOUR METHODS ##########################\n",
    "###########################################################################\n",
    "\n",
    "# Getting familiar with the functions I'll be using.\n",
    "# Symplectic method is giving me trouble...\n",
    "\n",
    "def analytic(y, t):\n",
    "    return 2*np.sin(t) + t\n",
    "\n",
    "def test_f(y, t):\n",
    "    return 2*np.cos(t) + 1\n",
    "\n",
    "def test_methods():\n",
    "    T = np.linspace(0,15,50)\n",
    "    y0 = [0]\n",
    "    \n",
    "    Y = analytic(y0, T)\n",
    "    \n",
    "    Y1 = euler_explicit(test_f, y0, T)\n",
    "    #Y2 = euler_symplectic(test_f, y0, T)\n",
    "    Y3 = rk2(test_f, y0, T)\n",
    "    Y4 = rk4(test_f, y0, T)\n",
    "    \n",
    "    plt.plot(T, Y,  label='analytic solution')\n",
    "    plt.plot(T, Y1, label='euler_explicit')\n",
    "    plt.title('euler_explicit method')\n",
    "    plt.xlabel('t')\n",
    "    plt.ylabel('y = sin(t) + t')\n",
    "    plt.legend(loc='best')\n",
    "    plt.show()\n",
    "    \n",
    "    #plt.plot(T, Y,  label='analytic solution')\n",
    "    #plt.plot(T, Y2, label='euler_symplectic')\n",
    "    #plt.title('euler_symplectic method')\n",
    "    #plt.xlabel('t')\n",
    "    #plt.ylabel('y = sin(t) + t')\n",
    "    #plt.legend(loc='best')\n",
    "    #plt.show()\n",
    "    \n",
    "    plt.plot(T, Y,  label='analytic solution')\n",
    "    plt.plot(T, Y3, label='rk2')\n",
    "    plt.title('rk2 method')\n",
    "    plt.xlabel('t')\n",
    "    plt.ylabel('y = sin(t) + t')\n",
    "    plt.legend(loc='best')\n",
    "    plt.show()\n",
    "    \n",
    "    plt.plot(T, Y,  label='analytic solution')\n",
    "    plt.plot(T, Y4, label='rk4')\n",
    "    plt.title('rk4 method')\n",
    "    plt.xlabel('t')\n",
    "    plt.ylabel('y = sin(t) + t')\n",
    "    plt.legend(loc='best')\n",
    "    plt.show()\n",
    "    \n",
    "test_methods()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EULER EXPLICIT INFO:\n",
      "dt: 0.020449061388082286\n",
      "Max height: 128.0627591939017\n",
      "y0:  [0, 50]\n",
      "EULER SYMPLECTIC INFO\n",
      "dt: 0.020449061388082286\n",
      "Max height: 0.0\n"
     ]
    },
    {
     "ename": "IndexError",
     "evalue": "too many indices for array",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mIndexError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-5-9aea1410eba1>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[0;32m    118\u001b[0m     \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    119\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 120\u001b[1;33m \u001b[0mno_drag\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m500\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    121\u001b[0m \u001b[0mdrag\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m500\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    122\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-5-9aea1410eba1>\u001b[0m in \u001b[0;36mno_drag\u001b[1;34m(N)\u001b[0m\n\u001b[0;32m     48\u001b[0m     \u001b[1;31m#r4 = rk4(f1, initial, t)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     49\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 50\u001b[1;33m     \u001b[0my1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv1\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mr1\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mr1\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     51\u001b[0m     \u001b[0my2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv2\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mr2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mr2\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     52\u001b[0m     \u001b[1;31m#y3, v3 = r3[:,0], r3[:,1]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mIndexError\u001b[0m: too many indices for array"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "def force_gravity(m=0.145, g=-9.8):\n",
    "    return m*g\n",
    "\n",
    "def force_drag(v, p=1.16, C_d=0.5, A=np.pi*0.0366**2):\n",
    "    # We need this to avoid division by zero\n",
    "    if v == 0:\n",
    "        return 0\n",
    "    \n",
    "    # v/abs(v) will be +1 or -1 depending on the direction of v\n",
    "    return (1/2*p*C_d*A * v**2) * v/abs(v)\n",
    "\n",
    "def acceleration(v, m=0.145, g=-9.8, p=1.16, C_d=0.5, A=np.pi*0.0366**2):\n",
    "    # a = F/m\n",
    "    return (force_gravity(m, g) + force_drag(v, p, C_d, A)) / m\n",
    "\n",
    "###########################################################################\n",
    "\n",
    "# Part (a)\n",
    "\n",
    "# dy/dt = v\n",
    "# dv/dt = g - (1/2*m)*p*C_d*A * v**2 * abs(v)/v\n",
    "\n",
    "# Part (b)\n",
    "# rk2 and rk4 methods aren't cooperating...\n",
    "\n",
    "def f1(y, t):\n",
    "    g = -9.8\n",
    "    y, v, = y[0], y[1]\n",
    "    return np.array([v, g])\n",
    "\n",
    "def f2(y, t):\n",
    "    g = -9.8\n",
    "    y, v = y[0], y[1]\n",
    "    return np.array([v, acceleration(v)])\n",
    "\n",
    "def no_drag(N):\n",
    "    y0, v0 = 0, 50\n",
    "    initial = [y0, v0]\n",
    "    \n",
    "    t_ground = 2*v0/9.8\n",
    "    t = np.linspace(0, t_ground, N)\n",
    "    \n",
    "    r1 = euler_explicit(f1, initial, t)\n",
    "    r2 = euler_symplectic(f1, initial, t)\n",
    "    #r3 = rk2(f1, initial, t)\n",
    "    #r4 = rk4(f1, initial, t)\n",
    "\n",
    "    y1, v1 = r1[:,0], r1[:,1]\n",
    "    y2, v2 = r2[:,0], r2[:,1]\n",
    "    #y3, v3 = r3[:,0], r3[:,1]\n",
    "    #y4, v4 = r4[:,0], r4[:,1]\n",
    "    \n",
    "    t_max = v0/9.8\n",
    "    analytic_max_height = -4.9*t_max**2 + v0*t_max\n",
    "    print('Analytic max height:', analytic_max_height)\n",
    "    \n",
    "    plt.plot(t, y1, label='euler_explicit')\n",
    "    plt.plot(t, y2, label='euler_symplectic')\n",
    "    #plt.plot(t, y3, label='rk2')\n",
    "    #plt.plot(t, y4, label='rk4')\n",
    "    plt.legend(loc='best')\n",
    "    plt.title('No drag height vs. time')\n",
    "    plt.xlabel('time (s)')\n",
    "    plt.ylabel('height (m)')\n",
    "    plt.show()\n",
    "    \n",
    "    plt.plot(t, v1, label='euler_explicit')\n",
    "    plt.plot(t, v2, label='euler_symplectic')\n",
    "    #plt.plot(t, y3, label='rk2')\n",
    "    #plt.plot(t, y4, label='rk4')\n",
    "    plt.legend(loc='best')\n",
    "    plt.title('No drag velocity vs. time')\n",
    "    plt.xlabel('time (s)')\n",
    "    plt.ylabel('velocity (m/s)')\n",
    "    plt.show()\n",
    "    \n",
    "def drag(N):\n",
    "    y0, v0 = 0, 50\n",
    "    initial = [y0, v0]\n",
    "    \n",
    "    t_ground = 2*v0/9.8\n",
    "    t = np.linspace(0, t_ground, N)\n",
    "    \n",
    "    r1 = euler_explicit(f2, initial, t)\n",
    "    r2 = euler_symplectic(f2, initial, t)\n",
    "    r3 = rk2(f1, initial, t)\n",
    "    #r4 = rk4(f1, initial, t)\n",
    "\n",
    "    y1, v1 = r1[:,0], r1[:,1]\n",
    "    y2, v2 = r2[:,0], r2[:,1]\n",
    "    y3, v3 = r3[:,0], r3[:,1]\n",
    "    #y4, v4 = r4[:,0], r4[:,1]\n",
    "    \n",
    "    t_max = v0/9.8\n",
    "    analytic_max_height = -4.9*t_max**2 + v0*t_max\n",
    "    print('Analytic max height:', analytic_max_height)\n",
    "    \n",
    "    plt.plot(t, y1, label='euler_explicit')\n",
    "    plt.plot(t, y2, label='euler_symplectic')\n",
    "    plt.plot(t, y3, label='rk2')\n",
    "    #plt.plot(t, y4, label='rk4')\n",
    "    plt.legend(loc='best')\n",
    "    plt.title('Drag')\n",
    "    plt.xlabel('time')\n",
    "    plt.ylabel('height (m)')\n",
    "    plt.show()\n",
    "    \n",
    "    plt.plot(t, v1, label='euler_explicit')\n",
    "    plt.plot(t, v2, label='euler_symplectic')\n",
    "    #plt.plot(t, y3, label='rk2')\n",
    "    #plt.plot(t, y4, label='rk4')\n",
    "    plt.legend(loc='best')\n",
    "    plt.title('No drag')\n",
    "    plt.xlabel('time (s)')\n",
    "    plt.ylabel('velocity (m/s)')\n",
    "    plt.show()\n",
    "\n",
    "no_drag(500)\n",
    "drag(500)\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "###########################################################################\n",
    "\n",
    "# From humble beginnings...just my initial attempts at this problem\n",
    "def net_force(v, m=0.145, g=-9.8, p=1.16, C_d=0.5, A=np.pi*0.0366**2):\n",
    "    # Let the +y direction be positive (rising) and -y direction be negative (falling)\n",
    "    \n",
    "    # If v > 0, then the drag force is in the same direction as gravity. If v < 0, it's in the opposite direction.\n",
    "    # When v = 0, the drag force disappears, so it doesn't matter if we tie that case in with v > 0 or v < 0.\n",
    "    if v >= 0:\n",
    "        k = 1\n",
    "    else:\n",
    "        k = -1\n",
    "        \n",
    "    force = (m*g) - k*(1/2*p*C_d*A * v**2)\n",
    "    \n",
    "    return force\n",
    "\n",
    "#net_force(0)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [],
   "source": [
    "C_d = 0\n",
    "rho = 1.16\n",
    "m = 0.145\n",
    "r = 0.0366\n",
    "g = 9.8\n",
    "A = np.pi*0.0366**2\n",
    "\n",
    "def accerlation(v,C_d=0.5, p=1.16, m=0.145, r=0.0366, g=-9.8, A=np.pi*0.0366**2):\n",
    "    return (force_gravity(m, g) + force_drag(v, p, A, C_d))/m\n",
    "\n",
    "def force_gravity(m=0.145,g=-9.8):\n",
    "    return m*g \n",
    "\n",
    "def force_drag(v, p=1.16, A=np.pi*0.0366**2, C_d=0.5):\n",
    "    #to avoid division by 0\n",
    "    if v==0:\n",
    "        return 0\n",
    "    else:\n",
    "        return -0.5 * p * v**2 * C_d * A / np.abs(v)\n",
    "\n",
    "def f1(y, t):\n",
    "    g = -9.8\n",
    "    y, v = y[0], y[1]\n",
    "    return np.array([v, g])\n",
    "    \n",
    "def f2(y, t):\n",
    "    g = -9.8\n",
    "    y, v = y[0], y[1]\n",
    "    return np.array([v, accelration(v)])\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EULER EXPLICIT INFO:\n",
      "dt: 0.0204490613881\n",
      "Max height: 128.062759194\n",
      "EULER EXPLICIT INFO:\n",
      "dt: 0.0204490613881\n",
      "Max height: 128.062759194\n",
      "Analytic max height: 127.55102040816327\n"
     ]
    },
    {
     "data": {
      "image/png": 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Ukb4iMhs76ki3qbN+4/fGTXi5am0rFCbg9brzTgjNw/HjC1i3I8bpOCYdUv3W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\nY+OuaSScT3Q6jk8K2GIx5MM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NsjuuMV7LioUJKClNq/rggw+ybds2Vq1axerVq3nooYfo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\njDHGpMmKhTHGmDT9P/9duWby3mQsAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c8105c60b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EULER EXPLICIT INFO:\n",
      "dt: 0.0204490613881\n",
      "Max height: 68.2742887721\n",
      "EULER EXPLICIT INFO:\n",
      "dt: 0.0204490613881\n",
      "Max height: 68.2742887721\n"
     ]
    },
    {
     "data": {
      "image/png": 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b8niNnf5rjO+8+sEMxhbJw8tJ+el017+djmO8yGfXkYhIWaAiHmNzqapfT/Rs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\nnj2bl156iaVLlwJw5513snb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6towxxmQzKyTGGGOyxAqJMcaYLLFCYowxJkuskBhjjMkS\nKyTGGGOyxAqJMcaYLLFCYkw6uOcR6eNxv7SIfO6jfbUTkRFpPF5LRD7wxb6NyQy7INGYdHCPtvyt\nqtbOhn39CNyhqkfSWGch8KCq7vZ1HmOuxFokxqTP80Bl9/DvL7jHc1sHICJdReRr98x520Wkn3u2\nwNUi8pN7ZFxEpLKIzBORX0RkmceYWxeJSDXg7IUiIiL3umfD/F1EYjxWnUPOnzjKBAgrJMakzxBg\nm6rWV9ULUzh7NuevwTXJUUPgWeCke6j4FcCFgSenAI+oagNcA+O9lcJ+mgCrPe4/DbRU1WuBth7L\nfwWaZu0lGeMdATVoozEOWqqqp4HTInIMV4sBYB1QW0QK4BpWfaZ7dFyA0BS2Uxr4x+P+D8BH7uMx\nX3ksP4hr3g9jHGeFxBjv8BxyWz3uJ+H6/ywIOOpupaTlDFDw4oZU+4pIA6AN8JuI1FfVo0A+97rG\nOM66toxJnxNARGafrKongL9F5J4Ly0SkTgqrbgSqeaxTWVV/UdVRuFoh5d0PVQf+zGweY7zJCokx\n6eA++P2jiKwVkReutHoqyzsBPURkjYj8yaXHPC5YDtTzuP+ie59rgZ9Uda17+c3Adxl4Ccb4jJ3+\na4yfEZGXcZ1qvCSVx/MAMcCNqpqUndmMSYm1SIzxP88B+dN4vAIwxIqI8RfWIjHGGJMl1iIxxhiT\nJVZIjDHGZIkVEmOMMVlihcQYY0yWWCExxhiTJf8HQSKSohAYRZsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c8110d4cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "NEXT PLOT IS FOR V0 = 500\n"
     ]
    },
    {
     "data": {
      "image/png": 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KJIltzaz0mg5ZNz7ipZfg0EOhceO4I3Eu+zRoEJZC3XlnOOoo+PbbuCNy6VKZ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\nzjmXlCcJ55xzSf1/5OrClC0fAvoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c810acae48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def no_drag_plot(N):\n",
    "    y0 = 0\n",
    "    v0 = 50\n",
    "    initial = [y0, v0]\n",
    "    t_ground = 2*v0/9.8\n",
    "    t = np.linspace(0, t_ground, N)\n",
    "\n",
    "    y1 = euler_explicit(f1, initial, t)\n",
    "    plt.plot(t, y1, label='euler')\n",
    "\n",
    "    y2 = euler_explicit(f1, initial, t)\n",
    "    plt.plot(t, y2, label='euler symplectic')\n",
    "\n",
    "    y3 = rk2(f1, initial, t)\n",
    "    plt.plot(t, y3, label='rk2')\n",
    "\n",
    "    y4 = rk4(f1, initial, t)\n",
    "    plt.plot(t, y4, label='rk4')\n",
    "    \n",
    "    t_max = v0/9.8\n",
    "    analytic_max_height = -4.9*t_max**2 + v0*t_max\n",
    "    print('Analytic max height:', analytic_max_height)\n",
    "    \n",
    "    plt.xlabel('time (s)')\n",
    "    plt.ylabel('height (m)')\n",
    "    plt.legend(loc='best')\n",
    "    plt.title('No Drag height vs time')\n",
    "    plt.show()\n",
    "    \n",
    "def drag_plot(N, v0, last_part=False): \n",
    "    y0 = 0\n",
    "    v0 = v0\n",
    "    initial = [y0, v0]\n",
    "    t_ground = 2*v0/9.8\n",
    "    \n",
    "    if last_part:\n",
    "        t_ground = 18\n",
    "    \n",
    "    t = np.linspace(0, t_ground, N)\n",
    "    \n",
    "    \n",
    "    if not last_part:\n",
    "        y1 = euler_explicit(f2, initial, t)\n",
    "        plt.plot(t, y1, label='euler')\n",
    "\n",
    "        y2 = euler_explicit(f2, initial, t)\n",
    "        plt.plot(t, y2, label='euler symplectic')\n",
    "\n",
    "        y3 = rk2(f2, initial, t)\n",
    "        plt.plot(t, y3, label='rk2')\n",
    "\n",
    "    y4 = rk4(f2, initial, t)\n",
    "    plt.plot(t, y4, label='rk4')\n",
    "\n",
    "    plt.xlabel('time (s)')\n",
    "    plt.ylabel('height (m)')\n",
    "    plt.legend(loc='best')\n",
    "    plt.title('Drag height vs time')\n",
    "    \n",
    "    if last_part:\n",
    "        plt.ylim((0,400))\n",
    "        \n",
    "    plt.show()\n",
    "\n",
    "no_drag_plot(500)\n",
    "drag_plot(500, 50)\n",
    "print('NEXT PLOT IS FOR V0 = 500')\n",
    "drag_plot(500, 500, last_part=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "N = 500 is actually good enough for ~1% error\n",
    "\n",
    "The max height calculated with Euler's method is slightly greater than the actual max height. This is expected though, since Euler's method uses linear approximations. Since the function is concave down, the peak will be overestimated.\n",
    "\n",
    "The motion depicted in the graph is a concave down parabola. The ball\n",
    "is constant accelerating downwards. With drag, the ball slows down much more quickly on the way up, and then accelerates more slowly on the way down until it reaches a terminal velocity.\n",
    "\n",
    "The magnitude of the ball's terminal velocity is about -33 m/s. You can see this by simply looking at the graph when it begins to straighten out. There's a point around (10, 250) and another point around (16, 50). The slope between those lines is -33.33 m/s. You could also get this from the velocity data which is solved for in every method we have implemented."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 3: Error Dependence on Time Step Size\n",
    "In this problem, you will write a function to evaluate the dependence of the integration error on the step size in the ODE integration methods that we have discussed.\n",
    "\n",
    "Begin with the harmonic oscillator problem with $m=k=1$, i.e.\n",
    "$$\n",
    "\\frac{d^2x}{dt^2} = -x\n",
    "$$\n",
    "With initial conditions $x(0)=1.0$ and $v(0)=0$.\n",
    "\n",
    "Make three plots, each with one \"curve\" for each integration method. All plots should have the step size `dt` on the horizontal axis.\n",
    " * The first plot should show the maximum magnitude of the error in $x$ on the vertical axis,  \n",
    " * the second plot should shown the maximum magnitude of the error in $v$, and \n",
    " * the third should show the maximum magnitude of the error in the normalized energy, (the magnitude of the error in the normalized energy is $\\left| x^2+v^2 - 1 \\right|$).\n",
    "\n",
    "In each case, the maximum error should be the magnitude of the largest difference from the analytical result for each quantity over the whole integration time. The number of steps needed to reach that time will depend on `dt`. Be sure to include plot titles and axis labels on each plot. Plot the results of all methods on the same plots, and use different colors for different methods.\n",
    "\n",
    "Decide whether it is most appropriate to use linear, semi-log, or log-log plots, and set the axis scales correctly. Describe the results, and try to be as quantitative as possible about the dependence and relative accuracy of the methods. For nice plots, try a maximum `dt=1.0`, a minimum `dt=0.01`, with at least 50 steps in `dt` equally spaced on a log scale. Integrate until you reach a maximum ${\\rm time}=50$ s at each `dt`. This calculation may take a while.\n",
    "\n",
    "------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 288,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "THIS GRAPH IS FOR ERROR IN X\n"
     ]
    },
    {
     "data": {
      "image/png": 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/rWGr0jLF2sSUT2pikrgtXw633QbXXAOHHQaXX65bckvz15QmJlWWpcWrroax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\nghARkayUIEREJKv/BwVaZ0j/bx8cAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c810c34710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def euler_explicit_step(func, y, t, dt):\n",
    "    return y + dt * func(y, t)\n",
    "\n",
    "# This is the same as the function at the beginning of the homework, except\n",
    "# this one doesn't have all the special stuff I added to cater to problem 2\n",
    "def euler_explicit(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0  # Store initial values\n",
    "    \n",
    "    for i in range(0,len(t)-1):\n",
    "        dt = t[i+1]-t[i]\n",
    "        y[i+1,:] = euler_explicit_step(func, y[i,:], t[i], dt)\n",
    "        \n",
    "    return y\n",
    "\n",
    "def euler_symplectic(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0[0]  # Store initial values\n",
    "        \n",
    "    for i in range(0,len(t)-1):\n",
    "        dt = t[i+1]-t[i]\n",
    "        # Build up so that, in general, the jth variable is updated \n",
    "        #   with the i+1 position of the (j-1)th variable.\n",
    "        # Requires an extra function call for each variable\n",
    "        # Currently only defined for 2-D.\n",
    "        r1 = euler_symplectic_step(func, y[i  ,:], t[i], dt)\n",
    "        y[i+1,0] = r1[0]\n",
    "        y[i+1,1] = y[i,1]\n",
    "\n",
    "        r2 = euler_symplectic_step(func, y[i+1,:], t[i], dt)\n",
    "        y[i+1,1] = r2[1]\n",
    "    return y\n",
    "\n",
    "def rk2(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0\n",
    "    for i in range(0, len(t)-1):\n",
    "        dt = t[i+1] - t[i]\n",
    "        y[i+1,:] = rk2_step(func, y[i,:], t[i], dt)\n",
    "    return y\n",
    "\n",
    "def rk4(func, y0, t):\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeros((len(t), num_variables))\n",
    "    y[0,:] = y0\n",
    "    for i in range(0, len(t)-1):\n",
    "        dt = t[i+1] - t[i]\n",
    "        y[i+1,:] = rk4_step(func, y[i,:], t[i], dt)\n",
    "    return y\n",
    "\n",
    "def f3_analytic(t):\n",
    "    return np.cos(t)\n",
    "\n",
    "def f3(y, t, k=1):\n",
    "    x, v = y[0], y[1]\n",
    "    return np.array([v, -k*x])\n",
    "\n",
    "def errors(N, X, V, E):\n",
    "    x0, v0, = 1, 0\n",
    "    T = np.linspace(0, 10, N)\n",
    "    initial = [x0, v0]\n",
    "    \n",
    "    r1 = euler_explicit(f3, initial, T)\n",
    "    r2 = euler_symplectic(f3, initial, T)\n",
    "    r3 = rk2(f3, initial, T)\n",
    "    r4 = rk4(f3, initial, T)\n",
    "    \n",
    "    x1, v1 = r1[:,0], r1[:,1]\n",
    "    x2, v2 = r2[:,0], r2[:,1]\n",
    "    x3, v3 = r3[:,0], r3[:,1]\n",
    "    x4, v4 = r4[:,0], r4[:,1]\n",
    "    \n",
    "    x1_err = max(abs(x1-f3_analytic(T)))\n",
    "    x2_err = max(abs(x2-f3_analytic(T)))\n",
    "    x3_err = max(abs(x3-f3_analytic(T)))\n",
    "    x4_err = max(abs(x4-f3_analytic(T)))\n",
    "    \n",
    "    v1_err = max(abs(v1-f3_analytic(T)))\n",
    "    v2_err = max(abs(v2-f3_analytic(T)))\n",
    "    v3_err = max(abs(v3-f3_analytic(T)))\n",
    "    v4_err = max(abs(v4-f3_analytic(T)))\n",
    "    \n",
    "    if X:\n",
    "        return x1_err, x2_err, x3_err, x4_err\n",
    "    elif V:\n",
    "        return v1_err, v2_err, v3_err, v4_err\n",
    "    elif E:\n",
    "        return (x1_err, x2_err, x3_err, x4_err, \n",
    "                v1_err, v2_err, v3_err, v4_err)\n",
    "\n",
    "def plot_error(X, V, E):\n",
    "    N = [ ]\n",
    "    E1 = [ ]\n",
    "    E2 = [ ]\n",
    "    E3 = [ ]\n",
    "    E4 = [ ]\n",
    "    for n in range(50, 5000+1, 50):\n",
    "        e1, e2, e3, e4 = errors(n, X, V, E)\n",
    "        N.append(n)\n",
    "        E1.append(e1)\n",
    "        E2.append(e2)\n",
    "        E3.append(e3)\n",
    "        E4.append(e4)\n",
    "    dt = 50*np.ones(len(N))\n",
    "    dt = dt/N\n",
    "    plt.plot(dt, E1, label='euler explicit')\n",
    "    plt.plot(dt, E2, label='euler symplectic')\n",
    "    plt.plot(dt, E3, label='rk2')\n",
    "    plt.plot(dt, E4, label='rk4')\n",
    "    plt.title('Error From Four Methods vs dt')\n",
    "    plt.xlabel('dt')\n",
    "    plt.ylabel('error')\n",
    "\n",
    "print('THIS GRAPH IS FOR ERROR IN X')\n",
    "plot_error(X=True, V=False, E=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 289,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "THIS GRAPH IS FOR ERROR IN V\n"
     ]
    },
    {
     "data": {
      "image/png": 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LS2VXsrnzdeOxRFVsgjCzW4ETgMXuPqxAuQOA54DT3P2+PGXc3WlthR/+MCSE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\niIhILCUIERGJpQQhIiKx/j+eaIiU4ynxmQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c81131c5f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print('THIS GRAPH IS FOR ERROR IN V')\n",
    "plot_error(X=False, V=True, E=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 296,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "THIS GRAPH IS FOR ERROR IN ENERGY\n"
     ]
    },
    {
     "data": {
      "image/png": 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szxG/gl/yWnX1xvH3X/hC6L8/8UT132dTbV1tZn3PrQzqjh06trtbo/F63cq6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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c810903630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def energy_errors(N):\n",
    "    x0, v0, = 1, 0\n",
    "    T = np.linspace(0, 10, N)\n",
    "    initial = [x0, v0]\n",
    "    \n",
    "    r1 = euler_explicit(f3, initial, T)\n",
    "    r2 = euler_symplectic(f3, initial, T)\n",
    "    r3 = rk2(f3, initial, T)\n",
    "    r4 = rk4(f3, initial, T)\n",
    "    \n",
    "    x1, v1 = r1[:,0], r1[:,1]\n",
    "    x2, v2 = r2[:,0], r2[:,1]\n",
    "    x3, v3 = r3[:,0], r3[:,1]\n",
    "    x4, v4 = r4[:,0], r4[:,1]\n",
    "    \n",
    "    e1_err = max(x1**2 + v1**2 -1)\n",
    "    e2_err = max(x2**2 + v2**2 -1)\n",
    "    e3_err = max(x3**2 + v3**2 -1)\n",
    "    e4_err = max(x4**2 + v4**2 -1)\n",
    "    \n",
    "    return e1_err, e2_err, e3_err, e4_err\n",
    "\n",
    "\n",
    "def plot_energy():\n",
    "    N = [ ]\n",
    "    E1 = [ ]\n",
    "    E2 = [ ]\n",
    "    E3 = [ ]\n",
    "    E4 = [ ]\n",
    "    for n in range(50, 5000+1, 50):\n",
    "        e1, e2, e3, e4 = energy_errors(n)\n",
    "        N.append(n)\n",
    "        E1.append(e1)\n",
    "        E2.append(e2)\n",
    "        E3.append(e3)\n",
    "        E4.append(e4)\n",
    "    dt = 50*np.ones(len(N))\n",
    "    dt = dt/N\n",
    "    plt.plot(dt, E1, label='euler explicit')\n",
    "    plt.plot(dt, E2, label='euler symplectic')\n",
    "    plt.plot(dt, E3, label='rk2')\n",
    "    plt.plot(dt, E4, label='rk4')\n",
    "    plt.title('Error From Four Methods vs dt')\n",
    "    plt.xlabel('dt')\n",
    "    plt.ylabel('energy error')\n",
    "   \n",
    "print('THIS GRAPH IS FOR ERROR IN ENERGY')\n",
    "plot_energy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.005\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1c810b775f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def g(x, v, t):\n",
    "    return np.array([v, -x])\n",
    "\n",
    "def euler_explicit_2d_step(f, x, v, t, dt):\n",
    "    return np.array([x, v]) + dt * f(x, v, t)\n",
    "\n",
    "def euler_explicit_2d(f, x0, v0, t):\n",
    "\n",
    "    x = np.zeros_like(t)\n",
    "    v = np.zeros_like(t)\n",
    "    x[0] = x0   # Initial value\n",
    "    v[0] = v0   # Initial value\n",
    "    \n",
    "    for i in range(0,len(t)-1):\n",
    "        dt = t[i+1]-t[i]\n",
    "        x[i+1], v[i+1] = euler_explicit_2d_step(f, x[i], v[i], t[i], dt)\n",
    "\n",
    "    return x, v\n",
    "\n",
    "N = 1000\n",
    "#t = np.linspace(0, 100, N)\n",
    "\n",
    "x0 = 1\n",
    "v0 = 0\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "def error_plot():\n",
    "    dt = [ ]\n",
    "    x1 = [ ]\n",
    "    v1 = [ ]\n",
    "    N = 1000\n",
    "    \n",
    "    while N < 10000:\n",
    "        T = np.linspace(0, 50, N)\n",
    "        dt_step = T[1] - T[0]\n",
    "        \n",
    "        x, v = euler_explicit_2d(g, x0, v0, T)\n",
    "        x_error = max(abs(x - np.cos(T)))\n",
    "        v_error = max(abs(v - (-np.sin(T))))\n",
    "        \n",
    "        dt.append(dt_step)\n",
    "        x1.append(x_error)\n",
    "        v1.append(v_error)\n",
    "        \n",
    "        N *= 1.05\n",
    "        \n",
    "    x1 = np.array(x1)\n",
    "    v1 = np.array(v1)\n",
    "    \n",
    "    plt.plot(dt, x1)\n",
    "    \n",
    "error_plot()\n",
    "blah = np.linspace(0, 50, 10001)\n",
    "print(blah[1]-blah[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem 4: Anharmonic Oscillator\n",
    "In this problem, you will write a function which uses the four integration methods demonstrated in the lecture demonstration to solve the anharmonic oscillator problem with $m=k=1$, i.e.\n",
    "\n",
    "$$\n",
    "\\frac{d^2x}{dt^2} = -x^m\n",
    "$$\n",
    "\n",
    "where $m$ is an odd integer. The initial conditions are $x(0)=1.0$ and $v(0)=0$.\n",
    "\n",
    "a. What is the formula for the normalized energy of the anharmonic oscillator for an arbitrary choice of $m\\geq 1$ and initial conditions $x(0)=x_0$ and $v(0)=0$? The normalized energy should always remain $1$ if energy is perfectly conserved.\n",
    "\n",
    "b. Write a function which integrates the anharmonic oscillator equation and plots the results:\n",
    "  1. Implement the four integration methods demonstrated in the lecture, with the method selectable in the graphical user interface.  \n",
    "  2. For more speed, only show `1` out of every `N_update` points on your plots, since plotting is likely the slowest part of the calculation. Even calculations with $10,000$ steps should be reasonably fast if you choose `N_update` $=100$ or more.  \n",
    "  3. On the first plot, show the $x$ and $v$ results over time. On the second plot, show the phase space trajectory. On the third plot, show the error in the normalized energy, which you found in part (a). Be sure to include plot titles and axis labels on each plot, and use different colors for plots with multiple data sets.  \n",
    "\n",
    "Note that we do not have an analytical solution to the general case, but check that your code is correct by verifying that the harmonic case, $m=1$, gives the expected results (describe what you expect and what you see).\n",
    "\n",
    "c. Make these three plots for all four integration methods (`Euler`, `Euler_symplectic`, `RK2`, and `RK4`) for the mildly anharmonic case: $m=3$, $dt=0.01$, total time $t=50.0$, and `N_update` $=10$ (each method should have three plots, don't combine the different methods on the same plot). Compare the results and the energy errors for all integration methods. Be sure to include plot titles and axis labels on each plot, and use different colors for plots with multiple data sets.\n",
    "\n",
    "d. Make these three plots for all four integration methods for an extremely anharmonic case: $m=101$, $dt=0.01$, total time $t=100.0$, and `N_update` $=10$ (note that the Euler method may not be stable enough to make it through the whole time interval).  Compare the results and the energy errors for all integration methods. Based on the results, what is the physical behavior of this highly anharmonic oscillator (i.e. what other physical system does it behave like)?\n",
    "\n",
    "e. Compare the changes in the oscillator behavior as you change the initial amplitude, $x_0$.\n",
    " * How does it depend on the choice of $m$?\n",
    " * What is special about $m=1$?\n",
    " * Why?\n",
    "\n",
    "------"
   ]
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   "source": [
    "The normalized energy is the integral...so, \n",
    "E = k * x^(m+1)/(m+1)\n",
    "\n",
    "The solution for m = 1 should look like x = -cos(t), which is the same analytic solution from the previous problem.\n",
    "\n",
    "This problem has been confusing me all week..."
   ]
  },
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   "source": [
    "def f4(y, t, k=1):\n",
    "    x, v = y[0], y[1]\n",
    "    return np.array([v, -k*x])\n",
    "\n",
    "\n",
    "###########################################################################\n",
    "\n",
    "def rk4_step(func, y, t, dt):\n",
    "    k1 = dt * func(y, t)\n",
    "    k2 = dt * func(y + k1/2, t + dt/2)\n",
    "    k3 = dt * func(y + k2/2, t + dt/2)\n",
    "    k4 = dt * func(y + k3, t + dt)\n",
    "    result = (1/6) * (k1 + 2*k2 + 2*k3 + k4)\n",
    "    y_new = y + result\n",
    "    return y_new\n",
    "\n",
    "def rk4(func, y0, t)\n",
    "    num_variables = len(y0)\n",
    "    y = np.zeroes((len(t), num_variables))\n",
    "    y[0,:] = y0\n",
    "    \n",
    "    for i in range(0, len(t)-1):\n",
    "        dt = t[i+1] - t[i]\n",
    "        y[i+1,:] = y[i,:] + rk4_step(func, y[i,:], t[i], dt)\n",
    "        \n",
    "    return y"
   ]
  }
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