Solver.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "6517cdab-a3bf-4159-b37b-986a2144ef2d",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy\n",
    "\n",
    "# s = np.int64(input(\"Please enter the degree of predictor-corrector method you want to be calculated: \"))\n",
    "s = 1\n",
    "m = 1\n",
    "t = sympy.symbols('t')\n",
    "P = sympy.Function('P')(t, s, m)\n",
    "def P(t,s,m):\n",
    "    res = 1\n",
    "    for l in range(s):\n",
    "        if l!= m:\n",
    "            res*= (t+s-1-l)/(m-l)\n",
    "    return res\n",
    "Q = sympy.Function('Q')(t, s, m)\n",
    "def Q(t,s,m):\n",
    "    res = 1\n",
    "    # This loop could be from 1 to s or 0 to s-1, I chose the latter\n",
    "    for l in range(s):\n",
    "        if l!= m:\n",
    "            res*= (t+s-2-l)/(m-l)\n",
    "    return res\n",
    "\n",
    "# This Function Collects the Adams-Bashforth and Adams-Moulton coefficients\n",
    "def abm_coef_collector(s):\n",
    "    # 'p's are the polynomial interpolations of the predictor\n",
    "    p = sympy.symarray(('p'), shape=s)\n",
    "    b = sympy.symarray(('b'),shape=s)\n",
    "    # 'q's are the polynomial interpolations of the corrector\n",
    "    q = sympy.symarray(('q'), shape=s)\n",
    "    c = sympy.symarray(('c'),shape=s)\n",
    "    bnum = np.zeros(s).reshape((1,s))\n",
    "    bdenom = np.zeros(s).reshape((1,s))\n",
    "    cnum = np.zeros(s).reshape((1,s))\n",
    "    cdenom = np.zeros(s).reshape((1,s))\n",
    "    ab = np.zeros(s).reshape((1,s))\n",
    "    abm = np.zeros(s).reshape((1,s))\n",
    "    for m in range(s):\n",
    "        p[m]=P(t,s,m)\n",
    "        b[m]=sympy.Rational(sympy.integrate(p[m],[t,0,1]).simplify())\n",
    "        bnum[0,m] = b[m].numerator\n",
    "        bdenom[0,m] = b[m].denominator\n",
    "        q[m]=Q(t,s,m)\n",
    "        c[m]=sympy.Rational(sympy.integrate(q[m],[t,0,1]).simplify())\n",
    "        cnum[0,m] = c[m].numerator\n",
    "        cdenom[0,m] = c[m].denominator\n",
    "        ab[0,m] = 1.*np.int64(bnum[0,m])/np.int64(bdenom[0,m])\n",
    "        abm[0,m] = 1.*np.int64(cnum[0,m])/np.int64(cdenom[0,m])\n",
    "    return ab,abm,bnum,bdenom,cnum,cdenom\n",
    "\n",
    "global one_sixth\n",
    "one_sixth = 1./6\n",
    "# Kuramoto Dynamics RHS of the equation\n",
    "def func(Theta,w0,N,k):\n",
    "    theta = Theta.reshape((1,N))\n",
    "    res = np.ones_like(theta)*w0\n",
    "    for i in range(N):\n",
    "        res[0,i] += k*np.sum(np.sin(theta[0,:]-theta[0,i]),axis = 0)\n",
    "    return res\n",
    "\n",
    "def rk4_step(Theta,w0,N,k,dt):\n",
    "    rk1 = func(Theta,w0,N,k)\n",
    "    rk2 = func(Theta + 0.5*dt*rk1,w0,N,k)\n",
    "    rk3 = func(Theta + 0.5*dt*rk2,w0,N,k)\n",
    "    rk4 = func(Theta + dt*rk3,w0,N,k)\n",
    "    res = Theta + one_sixth*(rk1+2*(rk2+rk3)+rk4)\n",
    "    return res, rk1\n",
    "\n",
    "def ab_step(dt,Theta_dot_History, ab):\n",
    "    res = np.dot(ab,Theta_dot_History)*dt\n",
    "    return res\n",
    "\n",
    "def abm_step(dt,Theta_dot_History, ab, abm,s):\n",
    "    dummy = Theta_dot_History\n",
    "    for i in range(s-1):\n",
    "        dummy[i,:] = dummy[i+1,:]\n",
    "    dummy[s-1,:] = np.dot(ab,Theta_dot_History)*dt\n",
    "    res = np.dot(abm,dummy)*dt\n",
    "    return res\n",
    "\n",
    "def initiate_kuramoto(N,omega0,K,dt,s):    \n",
    "    # Initiating the phases as a splay with random fluctuation\n",
    "    Theta0 = np.arange(0., 2*np.pi, 2*np.pi/N).reshape((1,N))+np.random.uniform(0.,1.,N).reshape((1,N))*1.0e-010\n",
    "    Theta = np.zeros_like(Theta0)\n",
    "    Theta_dot = np.zeros_like(Theta0)\n",
    "    Theta_History = np.zeros((s,N))\n",
    "    Theta_dot_History = np.zeros((s,N))\n",
    "    #Creating the history\n",
    "    Theta_History[0,:] = Theta0[0,:]\n",
    "    k = 1.*N/K\n",
    "    for i in range(s-1):\n",
    "        Theta_History[i+1,:], Theta_dot_History[i,:] = rk4_step(Theta_History[i,:],omega0,N,k,dt)\n",
    "    Theta = Theta_History[s-1,:]\n",
    "    Theta_dot_History[s-1,:] = func(Theta,omega0,N,k)\n",
    "    Theta_dot = Theta_dot_History[s-1,:]\n",
    "    return Theta0,Theta,Theta_dot,Theta_History,Theta_dot_History,k\n",
    "\n",
    "def Kuramoto(Final_Time,N,omega0,K,step_num,method,s):\n",
    "    dt = 1./step_num\n",
    "    Number_of_Steps = np.int64(Final_Time*step_num)\n",
    "    Time = np.linspace(-(s-1)*dt,Final_Time,Number_of_Steps+s)\n",
    "    Re_R = np.zeros(Number_of_Steps+s)\n",
    "    Im_R = np.zeros(Number_of_Steps+s)\n",
    "    rho = np.zeros(Number_of_Steps+s)\n",
    "    global k\n",
    "    Theta0,Theta,Theta_dot,Theta_History,Theta_dot_History,k = initiate_kuramoto(N,omega0,K,dt,s)\n",
    "    for i in range(s):\n",
    "        Re_R[i] = np.sum(np.cos(Theta_History[i,:]),axis=0)/N\n",
    "        Im_R[i] = np.sum(np.sin(Theta_History[i,:]),axis=0)/N\n",
    "    if method==\"rk4\":\n",
    "        for i in range(Number_of_Steps):\n",
    "            Theta, Theta_dot = rk4_step(Theta,omega0,N,k,dt)\n",
    "            for j in range(s-1):\n",
    "                Theta_History[j,:] = Theta_History[j+1,:]\n",
    "                Theta_dot_History[j,:] = Theta_dot_History[j+1,:]\n",
    "            Theta_History[s-1,:] = Theta\n",
    "            Theta_dot_History[s-1,:] = Theta       \n",
    "            Re_R[i+s] = np.sum(np.cos(Theta_History[s-1,:]),axis=0)/N\n",
    "            Im_R[i+s] = np.sum(np.sin(Theta_History[s-1,:]),axis=0)/N            \n",
    "    elif method==\"ab\":\n",
    "        for i in range(Number_of_Steps):\n",
    "            Theta = ab_step(dt,Theta_dot_History)\n",
    "            Theta_dot = func(Theta,omega0,N,k)\n",
    "            for j in range(s-1):\n",
    "                Theta_History[j,:] = Theta_History[j+1,:]\n",
    "                Theta_dot_History[j,:] = Theta_dot_History[j+1,:]\n",
    "            Theta_History[s-1,:] = Theta\n",
    "            Theta_dot_History[s-1,:] = Theta       \n",
    "            Re_R[i+s] = np.sum(np.cos(Theta_History[s-1,:]),axis=0)/N\n",
    "            Im_R[i+s] = np.sum(np.sin(Theta_History[s-1,:]),axis=0)/N                    \n",
    "    elif method==\"abm\":\n",
    "        for i in range(Number_of_Steps):\n",
    "            Theta = abm_step(dt,Theta_dot_History)\n",
    "            Theta_dot = func(Theta,omega0,N,k)\n",
    "            for j in range(s-1):\n",
    "                Theta_History[j,:] = Theta_History[j+1,:]\n",
    "                Theta_dot_History[j,:] = Theta_dot_History[j+1,:]\n",
    "            Theta_History[s-1,:] = Theta\n",
    "            Theta_dot_History[s-1,:] = Theta          \n",
    "            Re_R[i+s] = np.sum(np.cos(Theta_History[s-1,:]),axis=0)/N\n",
    "            Im_R[i+s] = np.sum(np.sin(Theta_History[s-1,:]),axis=0)/N            \n",
    "    else:\n",
    "        print(\"Invalid Method!!!\")\n",
    "    rho = np.sqrt(np.multiply(Re_R,Re_R)+np.multiply(Im_R,Im_R))\n",
    "    # phi = np.tan(np.divide(Im_R,Re_R))\n",
    "    return rho,Re_R,Im_R,Time\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "f499c305-0627-4426-99b8-2ee2b7ab9a16",
   "metadata": {},
   "outputs": [],
   "source": [
    "ab,abm,bnum,bdenom,cnum,cdenom = abm_coef_collector(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "c1b7b4f8-05f2-4eaa-9804-f03fe3a3c5cc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1, 12)\n"
     ]
    }
   ],
   "source": [
    "rho,Re_R,Im_R,Time = Kuramoto(100.,12,np.pi/2,1.,100,\"rk4\",6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "043a3450-7639-45d4-9d0b-08c3cf5a9b32",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7eec911adea0>]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot(Time,rho)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "61361270-57ed-4254-8e0f-0d6813e5a3e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho[-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e9432a0b-f8f4-46ad-9d8d-1818db73ea9f",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}