CoulombABM.ipynb

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   "source": [
    "### In this notebook, we implement the Adams-Bashforth-Milne time-stepper to simulate the 1-dimensional motion of two charged point particles accelerated due to the Coulomb interaction."
   ]
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   "source": [
    "#### Libraries"
   ]
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   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt"
   ]
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   "source": [
    "#### Just to get the code running, using the reaction\n",
    "$$^{236}U \\to ^{92}Kr + ^{141}Ba + 3n $$\n",
    "#### (ignoring the neutrons). For future simulations, we know fragment particle numbers will not be integers. We use this reaction simply to have not-totally-unrealistic parameters with which to test the performance of the solver.\n",
    "####\n",
    "#### Simulation Parameters:"
   ]
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    "# Masses and charges\n",
    "m1 = 92\n",
    "m1 *= 931.5 # Express mass in MeV\n",
    "q1 = 36\n",
    "m2 = 141\n",
    "m2 *= 931.5 # Express mass in MeV\n",
    "q2 = 56\n",
    "\n",
    "# Reduced mass\n",
    "mu = m1*m2/(m1+m2)\n",
    "\n",
    "# Coulomb constant (in MeV*fm)\n",
    "k = 1.44; \n",
    "\n",
    "# Initial conditions\n",
    "x01 = 15\n",
    "x02 = -15\n",
    "v01 = 0.05\n",
    "v02 = -m1*v01/m2\n",
    "\n",
    "# Energy\n",
    "E = m1*(v01)**2/2+m2*(v02)**2/2+k*q1*q2/np.abs(x01-x02)\n",
    "\n",
    "# Step Size and simulation time (fm/c). Number of steps.\n",
    "dt = 0.06\n",
    "T = 100\n",
    "s = int(np.ceil(T/dt))"
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   "source": [
    "#### From energy conservation, it follows that the system obeys the differential equation \n",
    "\n",
    "$$\\dot{x} = \\left[ \\frac{2}{\\mu}\\left(E-\\frac{k q_1 q_2}{|x|}\\right)\\right]^{1/2}$$\n",
    "\n",
    "#### We define a function to evaluate the RHS."
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   "source": [
    "def deriv(x):\n",
    "    return np.sqrt((2/mu)*(E-k*q1*q2/np.abs(x)))"
   ]
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   "id": "827a2e2a-171b-4630-b9b4-8843dbcd07be",
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   "source": [
    "#### To test the integration, compute a solution using the Forward Euler method."
   ]
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   "execution_count": 4,
   "id": "1cbb94c2-cb4f-4f9e-93a0-28ec91ee22ca",
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    "tags": []
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   "source": [
    "# Data storage arrays for Forward Euler integration\n",
    "xrelFE = np.zeros(s)\n",
    "xrelFE[0] = x01-x02\n",
    "\n",
    "# Forward Euler Time Stepper\n",
    "for t in range(0,s-1):\n",
    "    xrelFE[t+1] = xrelFE[t]+deriv(xrelFE[t])*dt;"
   ]
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  {
   "cell_type": "markdown",
   "id": "f303c94b-f9bb-40e5-97f0-5aa986815762",
   "metadata": {},
   "source": [
    "#### Plot Forward Euler test case trajectories and energy conservation"
   ]
  },
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   "execution_count": 5,
   "id": "059cbe73-a77d-4dff-9b9e-ef7fac84ed70",
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   ],
   "source": [
    "# Plotting configuration\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "plt.rcParams['figure.figsize'] = [8, 6]\n",
    "\n",
    "# Plot Forward Euler trajectories\n",
    "plt.plot([dt*i for i in range(s)],xrelFE+x02,c=\"b\",lw=2)\n",
    "plt.plot([dt*i for i in range(s)],-(xrelFE-x01),c=\"r\",lw=2)\n",
    "plt.xlabel(\"Time (fm/c)\")\n",
    "plt.ylabel(\"Position (fm)\")\n",
    "plt.title(\"Trajectories of Repelled Point Particles (Forward Euler)\")\n",
    "plt.show()"
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  },
  {
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   "execution_count": 6,
   "id": "2f43a63b-d86d-4fd4-a15a-69a8ae4f73b3",
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    "plt.plot([dt*i for i in range(s)],(E-[(1/2)*mu*deriv(x)**2 + k*q1*q2/np.abs(x) for x in xrelFE])/E,c=\"b\",lw=2)\n",
    "plt.xlabel(\"Time (fm/c)\")\n",
    "plt.ylabel(\"$\\Delta E/E$\")\n",
    "plt.title(\"Test Energy Conservation (Forward Euler)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34bceae7-bcec-4c5f-a7c7-5d7bfe51e8bb",
   "metadata": {},
   "source": [
    "#### Implement Adams-Bashforth-Milne time-stepper"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "d05e5ba5-7276-434e-8d10-f28b87e70cb2",
   "metadata": {},
   "outputs": [],
   "source": [
    "xrelABM = np.zeros(s);\n",
    "xrelABM[0:4] = xrelFE[0:4]; # One way to set up ICs.\n",
    "#xrelABM(1:4)=[30, 30.004929363986545,30.01006178186523, 30.01539718418845];\n",
    "\n",
    "ps = np.zeros(s); #WARNING: these start at 0. Arguably they should be initialized in some other way?\n",
    "ms = np.zeros(s);\n",
    "cs = np.zeros(s);\n",
    "for t in range(3,s-1):\n",
    "\n",
    "    # Predictor\n",
    "    ps[t+1] = (xrelABM[t] + xrelABM[t-1])/2 + (dt/48)*(119*deriv(xrelABM[t]) - \\\n",
    "                     99*deriv(xrelABM[t-1]) + 69*deriv(xrelABM[t-2]) - 17*deriv(xrelABM[t-3]) );\n",
    "        \n",
    "    # Modifier \n",
    "    ms[t+1] = ps[t+1]-(161/170)*(ps[t]-cs[t]);\n",
    "    \n",
    "    # Corrector\n",
    "    cs[t+1] = (xrelABM[t] + xrelABM[t-1])/2 + (dt/48)*(17*deriv(ms[t+1]) + \\\n",
    "        51*deriv(xrelABM[t]) + 3*deriv(xrelABM[t-1]) + deriv(xrelABM[t-2]) );\n",
    "\n",
    "    xrelABM[t+1] = cs[t+1]+(9/170)*(ps[t+1]-cs[t+1]);\n",
    "\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c22a2fd-f1a9-4826-a61a-bfdad6fd8d7c",
   "metadata": {},
   "source": [
    "#### Plot ABM trajectories and energy conservation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "5d254978-da75-470b-9452-2918ec716a68",
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      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
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     "output_type": "display_data"
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   ],
   "source": [
    "plt.plot([dt*i for i in range(s)],xrelABM+x02,c=\"b\",lw=2)\n",
    "plt.plot([dt*i for i in range(s)],-(xrelABM-x01),c=\"r\",lw=2)\n",
    "plt.plot([dt*i for i in range(s)],xrelFE+x02,\"b--\",lw=2)\n",
    "plt.plot([dt*i for i in range(s)],-(xrelFE-x01),\"r--\",lw=2)\n",
    "plt.xlabel(\"Time (fm/c)\")\n",
    "plt.ylabel(\"Position (fm)\")\n",
    "plt.title(\"Trajectories of Repelled Point Particles (ABM)\")\n",
    "plt.show()\n",
    "\n",
    "# Note that Forward Euler trajectories are also plotted, but are indistinguishable from ABM at this scale"
   ]
  },
  {
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   "execution_count": 18,
   "id": "0d0714ea-16a7-4f28-92f8-61d28a5a5d4d",
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   "source": [
    "plt.plot([dt*i for i in range(s)],deriv(xrelABM),\"b\",lw=2)\n",
    "plt.xlabel(\"Time (fm/c)\")\n",
    "plt.ylabel(\"v/c\")\n",
    "plt.title(\"Velocity of Relative Coordinate for Point Particles (ABM)\")\n",
    "plt.show()"
   ]
  },
  {
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   "execution_count": 19,
   "id": "729cd06f-c7c1-465d-9efa-ecd5e77b389e",
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   "source": [
    "plt.plot(xrelABM,deriv(xrelABM),\"b\",lw=2)\n",
    "plt.xlabel(\"Separation (fm)\")\n",
    "plt.ylabel(\"v/c\")\n",
    "plt.title(\"Velocity vs. Distance of Relative Coordinate for Point Particles (ABM)\")\n",
    "plt.show()"
   ]
  },
  {
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   "execution_count": 9,
   "id": "ad8c06c1-a1f3-4782-9f89-89971d88416f",
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   "source": [
    "plt.plot([dt*i for i in range(s)],(E-[(1/2)*mu*deriv(x)**2 + k*q1*q2/np.abs(x) for x in xrelABM])/E,c=\"b\",lw=2)\n",
    "plt.xlabel(\"Time (fm/c)\")\n",
    "plt.ylabel(\"$\\Delta E/E$\")\n",
    "plt.title(\"Test Energy Conservation (ABM)\")\n",
    "plt.show()"
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    "#### We compare the performance of the two solvers using the largest fractional difference, defined as \n",
    "$$\\frac{\\text{max}_t|x_{FE}(t)-x_{ABM}(t)|}{\\text{max}_t |x_{ABM}(t)|}$$"
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   "execution_count": 10,
   "id": "ada49c4a-bed9-4f4d-a252-0e2b5c847ffc",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.880571006433767e-06"
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     "execution_count": 10,
     "metadata": {},
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   "source": [
    "np.max( np.abs(xrelFE-xrelABM) )/np.max( np.abs(xrelABM) )"
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   "cell_type": "markdown",
   "id": "290d3121-c24a-4ac6-bd3d-a986b59509f0",
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   "source": [
    "#### Using Mathematica, it is also possible to solve explicitly for $t(x)$. Plugging in the numbers specific to our problem, we get\n",
    "$$t(x) = 0.588153 \\sqrt{x} \\sqrt{273.786 x-2903.04}+103.19\n",
    "   \\tanh\n",
    "   ^{-1}\\left(\\sqrt{\\frac{x-10.6033}{x}}\\right)-349\n",
    "   .306$$\n",
    "   \n",
    "#### This expression can be numerically inverted to find $x(t)$ for all $t$ in [dt*i for i in range(s)]. \n",
    "\n",
    "#### We copy the results of this numerical inversion below, and call these numbers the \"analytical\" solution."
   ]
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   "cell_type": "code",
   "execution_count": 13,
   "id": "8160f6f9-ba26-4dca-9f9d-2b59298edc86",
   "metadata": {
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    "xrelanalytical=np.array([30., 30.004957558752633, 30.00991534134419, \n",
    " 30.01487334770072, 30.01983157774832, \n",
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    " 38.2753230512939, 38.28056534733072, \n",
    " 38.28580778088725, 38.29105035192584, \n",
    " 38.296293060408836, 38.30153590629862, \n",
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    " 38.34872769641017, 38.35397191430135, \n",
    " 38.35921626918634, 38.36446076102767, \n",
    " 38.36970538978793, 38.37495015542968, \n",
    " 38.3801950579155, 38.385440097208, \n",
    " 38.39068527326978, 38.39593058606349, \n",
    " 38.401176035551764, 38.40642162169727, \n",
    " 38.41166734446267, 38.41691320381065, \n",
    " 38.42215919970393, 38.427405332105195, \n",
    " 38.4326516009772, 38.43789800628269, \n",
    " 38.4431445479844, 38.44839122604513, \n",
    " 38.453638040427656, 38.458884991094784, \n",
    " 38.46413207800932, 38.469379301134104, \n",
    " 38.474626660431966, 38.47987415586579, \n",
    " 38.48512178739842, 38.49036955499277, \n",
    " 38.49561745861172, 38.5008654982182, \n",
    " 38.50611367377512, 38.51136198524546, \n",
    " 38.51661043259214, 38.52185901577815])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aca220b6-3dd0-404f-b78d-34be2e137288",
   "metadata": {},
   "source": [
    "#### We compute the relative error between the two solvers and the \"analytical\" solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "abf5e7b3-5a2f-4536-a7c4-e1ba154aa4ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.888772455863576e-06"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.max( np.abs(xrelFE-xrelanalytical) )/np.max( np.abs(xrelanalytical) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "24d572b3-627e-4f6e-88b6-39de39ce2361",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.715482101413781e-09"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.max( np.abs(xrelABM-xrelanalytical) )/np.max( np.abs(xrelanalytical) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0be57687-29c3-4891-8cb0-22aa04367603",
   "metadata": {},
   "source": [
    "#### The ABM solver appears to be more accurate by two orders of magnitude."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "93cb1b8e-6268-4bb4-a5a6-e6986a2ca919",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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