RK4.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "6DMOsByKk_M4"
   },
   "source": [
    "# Runge-Kutta 4th order method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "5V0rGYgmfuWa"
   },
   "outputs": [],
   "source": [
    "# This is a standard initalisation code - no need to change this\n",
    "# It loads basic fast array code, plot library and ODE solver\n",
    "import math\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import solve_ivp\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats={'retina','png'}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "DRK3WXOXf_1l"
   },
   "outputs": [],
   "source": [
    "def f(t, x):\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align*}\n",
    "K_1 &= h f(x_n,t_n)\\\\\n",
    "K_2 &= h f\\left(x_n+\\frac{1}{2} K_1,t_n+\\frac{h}{2}\\right)\\\\\n",
    "K_3 &= h f\\left(x_n+\\frac{1}{2} K_2,t_n+\\frac{h}{2}\\right)\\\\\n",
    "K_4 &= h f\\left(x_n+K_3,t_n+h\\right)\\\\\n",
    "x_{n+1} &= x_n+ \\frac{K_1}{6}+\\frac{K_2}{3}+\\frac{K_3}{3}+\\frac{K_4}{6}\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 71
    },
    "colab_type": "code",
    "id": "PVrszgEQhRGd",
    "outputId": "fe0fd4f0-fa07-4b80-9fb1-8643c6fe99c7"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ti= [0, 0.2, 0.4, 0.6000000000000001, 0.8, 1.0, 1.2, 1.4, 1.5999999999999999, 1.7999999999999998, 1.9999999999999998, 2.1999999999999997]\n",
      "xi= [1, 1.2214, 1.49181796, 1.8221064563440001, 2.2255208257785615, 2.718251136605935, 3.3200719382504893, 4.055135865379147, 4.9529429459740895, 6.049524514212752, 7.388889241659456, 9.02478931976286]\n"
     ]
    }
   ],
   "source": [
    "## initial time and position\n",
    "t = 0\n",
    "tmax = 2\n",
    "x = 1\n",
    "\n",
    "## step\n",
    "h = 0.2\n",
    "\n",
    "## Lists to store the solution, and store the initial point\n",
    "ti = [t]\n",
    "xi = [x]\n",
    "\n",
    "while t < tmax:\n",
    "    K1 = h*f(t, x)\n",
    "    K2 = h*f(t+h/2, x+K1/2)\n",
    "    K3 = h*f(t+h/2, x+K2/2)\n",
    "    K4 = h*f(t+h, x+K3)\n",
    "    x = x+K1/6+K2/3+K3/3+K4/6\n",
    "    t = t+h\n",
    "    xi.append(x)\n",
    "    ti.append(t)\n",
    "\n",
    "print(\"ti=\", ti)\n",
    "print(\"xi=\", xi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 325
    },
    "colab_type": "code",
    "id": "t_BuqP_Ri9lo",
    "outputId": "5d69a15c-5726-4ab1-ea50-137f0d1838db"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f72faa4bd68>"
      ]
     },
     "execution_count": 9,
     "metadata": {
      "tags": []
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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QAgAANEoIAQAAGiWEAAAAjRJCAACARgkhAABAo4QQAACgUUIIAADQKCEEAABolBACAAA0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\nQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 291,
       "width": 400
      },
      "tags": []
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "\n",
    "ax.scatter(ti, xi, label=\"RK4\")\n",
    "ax.set_xlabel(\"$t$\")\n",
    "fig.legend()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 308
    },
    "colab_type": "code",
    "id": "tjsP0CwGjWrp",
    "outputId": "90d89160-feb4-467d-9d0b-763ee7dedb2d"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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0fOlXgDHGI6mzpEzLstLtuBcAAADqtinLMip1nsNIz13ci6lY9VB1r/C5UlKI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\nD22YhM1IXAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "execution_count": 10,
     "metadata": {
      "image/png": {
       "height": 291,
       "width": 400
      },
      "tags": []
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xexact = np.exp(ti)\n",
    "\n",
    "ax.plot(ti, xexact, label=\"Exact solution\")\n",
    "fig.legend()\n",
    "fig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "8WV_Nr_pk6sU"
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "colab": {
   "include_colab_link": true,
   "name": "RK4.ipynb",
   "provenance": [],
   "version": "0.3.2"
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}