{ "cells": [ { "cell_type": "markdown", "id": "37467eca", "metadata": {}, "source": [ "## Le schéma d'Euler\n", "\n", "$$y'(t)= \\frac{dy}{dt} \\approx \\frac{y(t+h) - y(t)}{h}$$\n", "\n", "$$y(t+h) \\approx y(t) + h\\,f(t,y(t))$$\n", "\n", "_On génère une suite de valeurs $y_n$ aux instants_ $t_n = t_0 + n\\cdot h$\n", "\n", "le schéma s'écrit\n", "$$y_{n+1} = y_n + h\\cdot f(t_n,y_n)$$\n", "où $y_0$ est la valeur initiale $y(t_0)$\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "cdd0961a", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "#schéma d'Euler\n", "def euler(f, t0, y0, h, tfin):\n", " t=np.arange(t0,tfin+h,h) # la subdivision, python n'atteint pas la borne superieure :-(\n", " y = np.zeros(len(t)) # initialisation\n", " y[0] = y0\n", " for n in range(len(t)-1):\n", " y[n+1] = y[n] + h*f(t[n], y[n])\n", " return y\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "3fc14dc9", "metadata": {}, "outputs": [], "source": [ "import math\n", "# definition de la fonction dans l'EDO\n", "def f(t,y):\n", " return y" ] }, { "cell_type": "code", "execution_count": 3, "id": "d698d5b2", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "# tinitial, donnee initiale, tfinal\n", "t0,y0=0.0,1.0 \n", "def quadeuler( h=1.0, tfin = 4.0):\n", "# appel de la fonction\n", " y=euler(f, t0, y0, h, tfin)\n", " t=np.arange(t0,tfin+h,h)\n", " plt.plot(t,y,'-ob')\n", " z=y0*np.exp(t)\n", " plt.plot(t,z,'or')\n", " tt=np.linspace(t[0],t[-1],100)\n", " #pour avoir exactement le meme intervalle que y\n", " plt.plot(tt,y0*np.exp(tt),'-r')\n", " plt.legend(['schéma euler','valeurs exactes','solution exacte'])\n", "quadeuler() " ] }, { "cell_type": "markdown", "id": "3e127927", "metadata": {}, "source": [ "## On peut faire varier le pas de la subdivision" ] }, { "cell_type": "code", "execution_count": 4, "id": "fd6c16b8", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Widget Javascript not detected. It may not be installed or enabled properly. Reconnecting the current kernel may help.\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "60236c78f2804164af9ffb2c125a34c6" } }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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\n", 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from ipywidgets import interact\n", "import ipywidgets as widgets\n", "\n", "interact(quadeuler, h=(1/64, 1.0), tfin=(1.0,7.0));" ] }, { "cell_type": "markdown", "id": "e99be8ce", "metadata": {}, "source": [ "## Convergence quand le pas de temps $h\\to 0$\n", "On peut essayer avec des _pas de plus en plus petits._" ] }, { "cell_type": "code", "execution_count": 5, "id": "7699af49", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "tfin=4.0\n", "pas = [1, 1/2, 1/4, 1/8, 1/16, 1/32,1/64]\n", "for h in pas:\n", " y=euler(f, t0, y0, h, tfin)\n", " t=np.arange(t0,tfin+h,h)\n", " plt.plot(t,y,'-ob',markersize=2)\n", " tt=np.arange(t0,tfin+1e-1,1e-1)\n", " plt.plot(tt,y0*np.exp(tt),'-r')\n", " plt.legend(['schéma euler','solution exacte'])" ] }, { "cell_type": "markdown", "id": "b05b125d", "metadata": {}, "source": [ "## Calcul des erreurs globales quand le pas de temps $h\\to 0$" ] }, { "cell_type": "code", "execution_count": 6, "id": "44a48e92", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "pas 1 erreur 3.3891\n", "pas 0.5 erreur 2.3266\n", "pas 0.25 erreur 1.4286\n", "pas 0.125 erreur 0.8058\n", "pas 0.0625 erreur 0.4304\n", "pas 0.03125 erreur 0.2228\n", "pas 0.015625 erreur 0.1134\n" ] } ], "source": [ "tfin=2.0\n", "pas = [1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64]\n", "err=np.zeros(len(pas))\n", "for i, h in enumerate(pas):\n", " y=euler(f,t0,y0,h,tfin)\n", " t=np.arange(t0,tfin+h,h)\n", " z=np.exp(t)\n", " erreur= np.max(np.abs(y-z))\n", " err[i]=erreur\n", "for i, h in enumerate(pas):\n", " print('pas', pas[i],'erreur',round(err[i],4)) " ] }, { "cell_type": "markdown", "id": "584ade89", "metadata": {}, "source": [ "## On trace en log log " ] }, { "cell_type": "code", "execution_count": 7, "id": "85c6e5de", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.loglog(pas, err,'-or')\n", "plt.loglog(pas,pas,'-ob')\n", "plt.legend(['erreur','droite de pente 1'])\n", "plt.xlabel('log pas')\n", "plt.ylabel('log erreur')\n", "plt.title('convergence Euler')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "58739b3f", "metadata": {}, "source": [ "Le fait que la pente vaut approximativement 1 traduit que le schéma d'Euler est d'ordre 1:\n", "\n", "si $err(h) = C h + o(h)$ alors $\\ln err(h) = \\ln h + \\ln C + o(1)$\n", "\n", "En coordonnée log-log, on a bien une droite de pente 1." ] }, { "cell_type": "code", "execution_count": null, "id": "42c064df", "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 }