{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.core.display import HTML\n",
"css_file = './custom.css'\n",
"HTML(open(css_file, \"r\").read())"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Python version: 3.10.6 (main, Nov 14 2022, 16:10:14) [GCC 11.3.0]\n",
"Numpy version: 1.21.5\n",
"Last Updated On: 2023-02-08 16:52:11.819086\n"
]
}
],
"source": [
"import sys #only needed to determine Python version number\n",
"print(f'Python version: {sys.version}')\n",
"\n",
"import numpy as np #only needed to determine Numpy version number\n",
"print(f\"Numpy version: {np.__version__}\")\n",
"\n",
"from datetime import datetime\n",
"print('Last Updated On: ', datetime.now())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# M62_CM1 Introduction à l'approximation numérique d'EDO \n",
"\n",
"\n",
"La résolution formelle d’équations différentielles s’avère très compliquée et limitée: la plupart des problèmes ne peuvent être qu’approchés.\n",
"\n",
"On ne peut expliciter des solutions analytiques que pour des équations différentielles ordinaires très particulières. \n",
"Par exemple : \n",
"- dans certains cas, on ne peut exprimer la solution que sous forme implicite. \n",
" C'est le cas par exemple de l'EDO $y'(t)=\\dfrac{y(t)-t}{y(t)+t}$ dont les solutions vérifient la relation implicite \n",
"$$\n",
"\\frac{1}{2}\\ln(t^2+y^2(t))+\\arctan\\left( \\frac{y(t)}{t} \\right)=C,\n",
"$$\n",
" où $C$ est une constante arbitraire.\n",
"- dans d'autres cas, on ne parvient même pas à représenter la solution sous forme implicite. \n",
" C'est le cas par exemple de l'EDO $y'(t)=e^{-t^2}$ dont les solutions ne peuvent pas s'écrire comme composition de fonctions élémentaires.\n",
"\n",
"\n",
"Lorsqu’on ne connait pas de solution exacte à un problème de Cauchy qui admet une et une seule solution, on essaye d’en avoir une bonne approximation par des méthodes numériques."
]
},
{
"cell_type": "markdown",
"metadata": {
"toc": true
},
"source": [
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Position du problème\n",
"\n",
"
\n",
"\n",
"**Problème de Cauchy:** \n",
" \n",
"trouver une fonction $y \\colon I\\subset \\mathbb{R} \\to \\mathbb{R}$ définie sur un intervalle $I$ telle que\n",
"$$\n",
"\\begin{cases}\n",
"y'(t) = \\varphi(t,y(t)), &\\forall t \\in I=]t_0,T[,\\\\\n",
"y(t_0) = y_0,\n",
"\\end{cases}\n",
"$$\n",
"avec $y_0$ une valeur donnée et supposons que l'on ait montré l'existence et l'unicité d'une solution $y$ pour $t\\in I$.\n",
"\n",
"
\n",
"\n",
"L'idée la plus simple pour résoudre de manière approchée un problème de Cauchy est de discrétiser l'intervalle de temps avec un pas $h$ et d'approcher la dérivée temporelle sur chaque sous-intervalle de longieur $h$.\n",
"\n",
"Pour $h>0$ soit $t_n\\equiv t_0+nh$ avec $n=0,1,2,\\dots,N$ une suite de $N+1$ nœuds de $I$ induisant une discrétisation de $I$ en $N$ sous-intervalles $I_n=[t_n;t_{n+1}]$ chacun de longueur $h=\\frac{T-t_0}{N}>0$ (appelé le *pas de discrétisation*).\n",
"\n",
"Pour chaque nœud $t_n$, on cherche la valeur inconnue $u_n$ qui approche la valeur exacte $y_n\\equiv y(t_n)$. \n",
"- L'ensemble de $N+1$ valeurs $\\{t_0, t_1=t_0+h,\\dots , t_{N}=T \\}$ représente les points de la *discrétisation*. \n",
"- L'ensemble de $N+1$ valeurs $\\{y_0, y_1,\\dots , y_{N} \\}$ représente la *solution exacte discrète*. \n",
"- L'ensemble de $N+1$ valeurs $\\{u_0 = y_0, u_1,\\dots , u_{N} \\}$ représente la *solution numérique*. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Construction élémentaire des méthodes d'Euler explicite et implicite"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Une méthode classique, la **méthode d'Euler explicite** (ou *progressive*, de l'anglais *forward*), est obtenue en considérant l'équation différentielle en chaque nœud $t_n$ et en remplaçant la dérivée exacte $y'(t_n)$ par le taux d'accroissement\n",
"$$\n",
"\\varphi(t_n,y(t_n))=y'(t_n)=\\lim_{h\\to0}\\frac{y(t_n+h)-y(t_n)}{h}\\simeq\\frac{y(t_{n+1})-y(t_n)}{h}.\n",
"$$\n",
"Cela permet de construire une solution numérique par une suite récurrente:\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Ces deux méthodes sont dites *à un pas*: pour calculer la solution numérique $u_{n+1}$ au nœud $t_{n+1}$, on a seulement besoin des informations\n",
"disponibles au nœud précédent $t_n$.\n",
"Plus précisément, \n",
"- pour la méthode d'Euler progressive, $u_{n+1}$ ne dépend que de la valeur $u_n$ calculée précédemment, \n",
"- pour la méthode d'Euler rétrograde, $u_{n+1}$ dépend aussi \"de lui-même\" à travers la valeur de $\\varphi(t_{n+1},u_{n+1})$. \n",
"\n",
"C'est pour cette raison que la méthode d'Euler progressive est dite *explicite* tandis que la méthode d'Euler rétrograde est dite *implicite*. \n",
"Les méthodes implicites sont plus coûteuses que les méthodes explicites car, si la fonction $\\varphi$ est non linéaire, un problème non linéaire doit être résolu à chaque temps $t_{n+1}$ pour calculer $u_{n+1}$. \n",
"Néanmoins, nous verrons que les méthodes implicites jouissent de meilleures propriétés de stabilité que les méthodes explicites."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Implémentation des schémas d'Euler explicite et implicite\n",
"\n",
"Voyons un exemple complet.\n",
"\n",
"
\n",
"\n",
"**Exemple:** considérons le problème de Cauchy\n",
" \n",
">trouver la fonction $y \\colon I\\subset \\mathbb{R} \\to \\mathbb{R}$ définie sur l'intervalle $I=[0,1]$ telle que\n",
">$$\n",
"\\begin{cases}\n",
"y'(t) = 2ty(t), &\\forall t \\in I=[0,1],\\\\\n",
"y(0) = 1.\n",
"\\end{cases}\n",
"$$\n",
" \n",
"Sachant que la solution est $y(t)=e^{t^2}$, estimer la qualité du schéma en affichant la plus grande erreur commise (en valeur absolue). \n",
" \n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On commence par importer \n",
"1. la fonction `fsolve` du module `scipy.optimize` pour résoudre les équations implicites présentes dans le schéma implicite.\n",
"2. le module `matplotlib` et le module `numpy`:\n",
" - soit en important séparemment `matplotlib.pyplot` et `numpy` (avec les alias classiques);\n",
" - soit en important juste `matplotlib.pylab`, ce qui importe à la fois `matplotlib` et `numpy`. Dans ce cas on utilise la méthode du \"paresseau\" (i.e. on utilise `*`). \n",
" \n",
"Rappels: `numpy` rédéfinit toutes les fonctions mathématiques du module `math` et ces fonctions sont vectorisées (e.g. on pourra écrire directement `yy=sin(xx)` avec `xx` une liste/tuple/array au lieu d'écrire `yy=[sin(x) for x in xx]`)."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"%reset -f\n",
"%matplotlib inline\n",
"\n",
"from scipy.optimize import fsolve\n",
"\n",
"#from matplotlib.pylab import *\n",
"# rcdefaults()\n",
"# rc('font', size=16)\n",
"\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"# plt.rcdefaults()\n",
"plt.rcParams.update({'font.size': 16})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On initialise le problème de Cauchy"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"t0 = 0\n",
"tfinal = 1\n",
"y0 = 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On définit l'équation différentielle : `phi` est une fonction python qui contient la fonction mathématique $\\varphi\\colon [t_0,T]\\times\\mathbb{R}\\to\\mathbb{R}$ définie par $\\varphi(t, y)=2ty$. Bien noter que $\\varphi$ dépend de **deux** variables: $t$ et $y$. \n",
"Dans l'EDO on a $\\varphi(t,y(t))$: il s'agit d'une fonction d'**une** seule variable obtenue par composition de la fonction $\\varphi$ avec la fonction $t\\mapsto y(t)$."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"phi = lambda t,y : 2*y*t"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On introduit la discrétisation: les nœuds d'intégration $[t_0,t_1,\\dots,t_{N}]$ sont contenus dans le vecteur `tt`. \n",
"On a $N+1$ points espacés de $h=\\frac{t_N-t_0}{N}$."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"N = 8\n",
"tt = np.linspace(t0,tfinal,N+1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On écrit les schémas numériques : les valeurs $[u_0,u_1,\\dots,u_{N}]$ pour chaque méthode sont contenues dans le vecteur `uu`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Schéma d'Euler progressif :**\n",
"$$\\begin{cases}\n",
"u_0=y_0,\\\\\n",
"u_{n+1}=u_n+h\\varphi(t_n,u_n)& n=0,1,2,\\dots N-1\n",
"\\end{cases}$$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"def euler_progressif(phi,tt,y0):\n",
" h = tt[1]-tt[0]\n",
" N = len(tt)-1 # len(tt) = N+1 car tt contient N+1 points\n",
" uu = np.array([y0]) # array à partir d'une liste qui contient un seul element\n",
" for n in range(N): # range(N) = 0,1,...,N-1 \n",
" uu = np.append( uu, uu[n]+h*phi(tt[n],uu[n]) )\n",
" return uu\n",
"\n",
"# même fonction sans utiliser les array de numpy mais juste des listes\n",
"# def euler_progressif(phi,tt,y0):\n",
"# h = tt[1]-tt[0]\n",
"# uu = [y0] \n",
"# for n in range(len(tt)-1): \n",
"# uu.append( uu[-1]+h*phi(tt[-1],uu[-1]) )\n",
"# return uu # on renvoie une liste au lieu d'un np.array"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Schéma d'Euler régressif :**\n",
"$$\\begin{cases}\n",
"u_0=y_0,\\\\\n",
"u_{n+1}=u_n+h\\varphi(t_{n+1},u_{n+1})& n=0,1,2,\\dots N-1\n",
"\\end{cases}$$\n",
"\n",
"Attention : \n",
"- $u_{n+1}$ est solution de l'équation $x=u_n+h\\varphi(t_{n+1},x)$, c'est-à-dire un zéro de la fonction (en general non linéaire) $$x\\mapsto -x+u_n+h\\varphi(t_{n+1},x)$$\n",
"- la fonction `fsolve` du module `scipy.optimize` requiert deux paramètres : une fonction et un point de départ. Elle renvoie une liste avec la valeur du zéro approché (c'est une liste car `fsolve` resout un système, ici avec une seule équation)."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"def euler_regressif(phi,tt,y0):\n",
" h = tt[1]-tt[0]\n",
" uu = np.array([y0])\n",
" for n in range(len(tt)-1):\n",
" temp = fsolve( lambda x: -x+uu[n]+h*phi(tt[n+1],x) , uu[n] )\n",
" uu = np.append( uu, temp[0] )\n",
" return uu"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On calcule les solutions approchées:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"uu_ep = euler_progressif(phi,tt,y0)\n",
"uu_er = euler_regressif(phi,tt,y0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Comme on la connait, on définit la solution exacte pour calculer les erreurs:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"sol_exacte = lambda t : y0*np.exp(t**2)\n",
"yy = sol_exacte(tt) # = [sol_exacte(t) for t in tt] "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On compare les graphes des solutions exacte (en bleu) et approchées (en rouge) et on affiche le maximum de l'erreur:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(18, 7))\n",
"\n",
"plt.subplot(1,2,1)\n",
"plt.plot(tt,yy ,'b-o',label='Exacte')\n",
"plt.plot(tt,uu_ep,'r-D',label='Approchée')\n",
"erreur = np.abs(uu_ep-yy) # = [abs(uu_ep[i]-yy[i]) for i in range(N)] = [abs(u-y) for u,y in zip(uu_ep,yy)]\n",
"plt.title(f'Euler explicite\\nmax(|erreur|) = {max(erreur):g}') # idem que np.norm(uu_ep-yy,inf)\n",
"plt.grid()\n",
"plt.legend();\n",
"\n",
"plt.subplot(1,2,2)\n",
"plt.plot(tt,yy ,'b-o',label='Exacte')\n",
"plt.plot(tt,uu_er,'r-D',label='Approchée')\n",
"erreur = np.abs(uu_er-yy)\n",
"plt.title(f'Euler implicite\\nmax(|erreur|) = {max(erreur):g}') \n",
"plt.grid()\n",
"plt.legend();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Convergence théorique des schémas d'Euler\n",
"\n",
"\n",
"
\n",
" \n",
"**Définition: ordre de convergence** \n",
"Si $C(h) = \\mathcal{O}(h^p)$ pour $p > 0$, on dit que la convergence de la méthode est d'ordre $p$. \n",
"\n",
"
\n",
"\n",
"Remarque: $N\\to+\\infty$ lorsque $h\\to0$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Soit $u_{n+1}^*$ la solution numérique au temps $t_{n+1}$ qu'on obtiendrait en insérant de la solution exacte dans le schéma (par exemple, pour la méthode d'Euler explicite on a $u_{n+1}^*\\equiv y_{n} + h\\varphi(t_{n}, y_{n})$).\n",
"Pour vérifier qu'une méthode converge, on écrit l'erreur ainsi\n",
"\t\\begin{equation}\\label{quart7.9}\n",
"\te_n \\equiv y_n - u_n = (y_n - u_n^*) + (u_n^* - u_n).\n",
"\t\\end{equation}\n",
"Si les deux termes $(y_n - u_n^*)$ et $(u_n^* - u_n)$ tendent vers zéro quand $h \\to 0$ alors la méthode converge.\n",
"\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"
\n",
" \n",
"**Définition: erreur de troncature** \n",
"La quantité\n",
"$$\n",
"\\tau_{n+1}(h)\\equiv\\frac{y_{n+1}-u_{n+1}^*}{h}\n",
"$$\n",
"est appelée erreur de troncature locale. \n",
"Elle représente (à un facteur $1/h$ près) l'erreur qu'on obtient en insérant la solution exacte dans le schéma numérique. \n",
"\n",
"L'**erreur de troncature globale** (ou plus simplement l'erreur de troncature) est définie par\n",
"$$\n",
"\\tau(h)=\\max_{n=0,\\dots,N}|\\tau_n(h)|.\n",
"$$\n",
"\n",
"
\n",
"\n",
"
\n",
" \n",
"**Définition: consistance** \n",
"Si $\\lim_{h\\to0} \\tau (h) = 0$ on dit que la méthode est consistante. \n",
"\n",
"
\n",
"\n",
"
\n",
" \n",
"**Définition: ordre de consistance** \n",
"On dit qu'elle est consistante d'ordre $p$ si $\\tau (h) = \\mathcal{O}(h^p)$ pour un certain $p\\ge1$.\n",
"
\n",
"\n",
"\n",
"Remarque: la propriété de consistance est nécessaire pour avoir la convergence. \n",
"En effet, si elle n'était pas consistante, la méthode engendrerait à chaque itération une erreur qui ne tendrait pas vers zéro avec $h$. \n",
"L'accumulation de ces erreurs empêcherait l'erreur globale de tendre vers zéro quand $h \\to 0$. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"**Proposition.** \n",
"Si la fonction $\\varphi$ est lipschitzienne de constante $L$ par rapport à sa deuxième variable, la méthode d'Euler explicite appliquée au problème de Cauchy $y'(t)=\\varphi(t,y(y))$ pour $t\\in[t_0;T]$ et $y(t_0)=y_0$ est convergente d'ordre 1.\n",
"
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Preuve** \n",
">On étudie séparément l'erreur de consistance et l'accumulation de ces erreurs. On en déduit ensuite la convergence."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
">+ **Terme $y_n - u_n^*$** (erreur de consistance). \n",
" Il représente l'erreur engendrée par une seule itération de la méthode d'Euler explicite. \n",
" En supposant que la dérivée seconde de $y$ existe et est continue, on écrit le développement de Taylor de $y$ au voisinage de $t_n$:\n",
" $$\n",
" y(t_{n+1})=y(t_n)+hy'(t_n)+\\frac{h^2}{2}y''(\\eta_n)\n",
" $$\n",
" où $\\eta_n$ est un point de l'intervalle $]t_n;t_{n+1}[$. \n",
" Donc il existe $\\eta_n \\in ]t_n, t_{n+1}[$ tel que\n",
" $$\n",
" y_{n+1}-u_{n+1}^*\n",
" =y_{n+1}-\\Big(y_{n} + h\\varphi(t_{n}, y_{n})\\Big)=y_{n+1}-y_{n} - hy'(t_n)=\\frac{h^2}{2}y''(\\eta_n).\n",
" $$\n",
" L'erreur de troncature de la méthode d'Euler explicite est donc de la forme\n",
" $$\n",
" \\tau(h)=M\\frac{h}{2},\n",
" \\qquad\n",
" M\\equiv\\max_{t\\in [t_0,T]}|y''(t)|.\n",
" $$\n",
" On en déduit que $\\lim_{h\\to0} \\tau (h) = 0$: la méthode est consistante."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" \n",
">+ **Terme $u_{n+1}^* - u_{n+1}$** (accumulation des erreurs). \n",
" Il représente la propagation de $t_{n}$ à $t_{n+1}$ de l'erreur accumulée au temps précédent $t_{n}$.\n",
" On a \n",
"> \n",
">$$u_{n+1}^* - u_{n+1} =\n",
" \\Big(y_n + h \\varphi(t_n, y_n)\\Big)-\\Big(u_n + h \\varphi(t_n, y_n)\\Big)=e_n+h\\Big( \\varphi(t_n,y_n)-\\varphi(t_n,u_n) \\Big).\n",
" $$\n",
">\n",
">Comme $\\varphi$ est lipschitzienne par rapport à sa deuxième variable, on a\n",
" $$\n",
" |u_{n+1}^{*} - u_{n+1}|\\le (1 + hL)|e_{n}|.\n",
" $$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" \n",
">+ **Convergence.** \n",
" Comme $e_0 = 0$, les relations précédentes donnent\n",
" \\begin{align*}\n",
"\t|e_n| \n",
"\t&\\le\n",
"\t|y_n - u_n^*| + |u_n^* - u_n|\n",
"\t\\\\\n",
"\t&\\le\n",
"\th|\\tau_n(h)| + (1+hL)|e_{n-1}|\n",
"\t\\\\\n",
"\t&\\le\n",
"\th|\\tau_n(h)| + (1+hL)\\left( h|\\tau_{n-1}(h)| + (1+hL)|e_{n-2}| \\right)|\n",
"\t\\\\\n",
"\t&\\le\n",
"\t\\left( 1+(1+hL)+\\dots+(1+hL)^{n-1} \\right)h\\tau(h)\n",
"\t\\\\\n",
"\t&=\\left(\\sum_{i=0}^{n-1}(1+hL)^i\\right)h\\tau(h)\n",
"\t\\\\\n",
"\t&=\\frac{(1+hL)^n-1}{hL}h\\tau(h)\n",
"\t\\\\\n",
"\t&\\le\\frac{(e^{hL})^n-1}{hL}h\\tau(h) \\qquad\\text{car }(1+x)\\le e^x\n",
"\t\\\\\n",
"\t&=\\frac{(e^{hL})^{(t_n-t_0)/h}-1}{L}\\tau(h)\\qquad\\text{car } t_n-t_0=nh\n",
"\t\\\\\n",
"\t&=\\frac{e^{L(t_n-t_0)}-1}{L}\\tau(h)\n",
"\t\\\\\n",
"\t&=\\frac{e^{L(t_n-t_0)}-1}{L}\\frac{M}{2}h=\\mathcal{O}(h)\n",
"\\end{align*}\n",
"On peut conclure que la méthode d'Euler explicite est convergente d'ordre 1. \n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On remarque que l'ordre de cette méthode coïncide avec l'ordre de son erreur de troncature. \n",
"On retrouve cette propriété dans de nombreuses méthodes de résolution numérique d'équations différentielles ordinaires.\n",
" \n",
"\n",
"Remarque: l'estimation de convergence est obtenue en supposant seulement $\\varphi$ lipschitzienne. \n",
"\tOn peut établir une meilleure estimation si $\\partial_y\\varphi$ existe et est non négative pour tout $t \\in [t_0;T]$ et tout $y\\in\\mathbb{R}$. \n",
"\tEn effet dans ce cas \n",
"\t$$\\begin{align*}\n",
"\tu_n^* - u_n \n",
"\t&=\n",
"\t\\left(y_{n-1} + h\\varphi(t_{n-1}, y_{n-1})\\right)\t-\n",
"\t\\left( u_{n-1} + h\\varphi(t_{n-1},u_{n-1})\\right)\n",
"\t\\\\\n",
"\t&=\n",
"\te_{n-1}+h\\left( \\varphi(t_{n-1},y_{n-1})-\\varphi(t_{n-1},u_{n-1}) \\right)\n",
"\t\\\\\n",
"\t&=\n",
"\te_{n-1}+h\\left( e_{n-1}\\partial_y\\varphi(t_{n-1},\\eta_n) \\right)\n",
"\t\\\\\n",
"\t&=\n",
"\t\\left( 1+h\\partial_y\\varphi(t_{n-1},\\eta_n) \\right)e_{n-1}\n",
"\t\\end{align*}$$\n",
"\toù $\\eta_n$ appartient à l'intervalle dont les extrémités sont $y_{n-1}$ et $u_{n-1}$. \n",
"\tAinsi, si\n",
"\t$$\n",
"\t0"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(10, 7))\n",
"plt.loglog(H,err_ep, 'r-o',label='Euler Explicite')\n",
"plt.loglog(H,err_er, 'g-+',label='Euler Implicite')\n",
"plt.xlabel('$h$')\n",
"plt.ylabel('erreur')\n",
"plt.legend(bbox_to_anchor=(1.04,1),loc='upper left')\n",
"plt.grid(True);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Pour estimer l'ordre de convergence on doit estimer la pente de la droite qui relie l'erreur au pas $k$ à l'erreur au pas $k+1$ en échelle logarithmique.\n",
"Pour estimer la pente globale de cette droite (par des moindres carrés) on peut utiliser la fonction `polyfit` (du module `numpy` que nous avons déjà importé avec `matplotlib.pylab`). "
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Euler progressif: ordre 0.91\n",
"Euler regressif: ordre 1.13\n"
]
}
],
"source": [
"# ln(e) = a ln(h) + b\n",
"a_ep, b_ep = np.polyfit(np.log(H),np.log(err_ep), 1) # polyfit ( [liste des abscisses], [liste des ordonnées], degré du polynome)\n",
"print (f'Euler progressif: ordre {a_ep :1.2f}')\n",
"a_er, b_er = np.polyfit(np.log(H),np.log(err_er), 1)\n",
"print (f'Euler regressif: ordre {a_er:1.2f}')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On peut bien-sûr afficher la droite obtenue par régression linéaire en même temps que les points.\n",
"\n",
"Soit on affiche les logarithmes des données et l'équation de la droite avec la commande ``plot``:"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(18, 7))\n",
"\n",
"plt.subplot(1,2,1)\n",
"plt.plot(np.log(H),np.log(err_ep), 'ro',label='Euler Explicite')\n",
"plt.plot(np.log(H),[a_ep*np.log(h)+b_ep for h in H])\n",
"plt.xlabel('$\\ln(h)$')\n",
"plt.ylabel('$\\ln(e)$')\n",
"plt.legend(loc='upper left')\n",
"plt.grid(True);\n",
"\n",
"plt.subplot(1,2,2)\n",
"plt.plot(np.log(H),np.log(err_er), 'ro',label='Euler Implicite')\n",
"plt.plot(np.log(H),[a_er*np.log(h)+b_er for h in H])\n",
"plt.xlabel('$\\ln(h)$')\n",
"plt.ylabel('$\\ln(e)$')\n",
"plt.legend(loc='upper left')\n",
"plt.grid(True);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Soit on affiche les données en échelle logarithmique et l'équation de l'erreur $Ch^p$ avec $C=\\ln(b)$ et $p=a$ avec l'instruction ``loglog``:"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.figure(figsize=(18, 7))\n",
"\n",
"plt.subplot(1,2,1)\n",
"plt.loglog(H,err_ep, 'ro',label='Euler Explicite')\n",
"plt.loglog(H,[np.exp(b_ep)*(h**a_ep) for h in H])\n",
"plt.xlabel('$h$')\n",
"plt.ylabel('$e$')\n",
"plt.legend(loc='upper left')\n",
"plt.grid(True);\n",
"\n",
"plt.subplot(1,2,2)\n",
"plt.loglog(H,err_er, 'ro',label='Euler Implicite')\n",
"plt.loglog(H,[np.exp(b_er)*(h**a_er) for h in H])\n",
"plt.xlabel('$h$')\n",
"plt.ylabel('$e$')\n",
"plt.legend(loc='upper left')\n",
"plt.grid(True);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Annexe: implémentation d'un schéma \"moyenne\" des deux schémas précédents"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On obtient la **méthode de Crank-Nicolson** (ou *du trapèze*)\n",
"