Lecture 06.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finite difference method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "Nh = 20\n",
    "h = 1\n",
    "Nk = 21\n",
    "k = 1./Nk\n",
    "r = k/(h*h)\n",
    "x = np.linspace(0.,1.,Nk)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update(u):\n",
    "    new_u = np.zeros(len(u))\n",
    "    for i in range(1,len(u)-1):\n",
    "        new_u[i] = (1.-2.*r)*u[i] + r*u[i-1] + r*u[i+1]\n",
    "    return new_u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_init = np.zeros(Nk)\n",
    "u_init[Nk//2] = 1\n",
    "u = u_init\n",
    "for i in range(Nh):\n",
    "    u = update(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11169d198>]"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1)\n",
    "ax.plot(x,u_init)\n",
    "ax.plot(x,u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}