From 4fa55a69c8eb2f69fbf82282ab5c9106f9e8c5f2 Mon Sep 17 00:00:00 2001
From: gergaud <joseph.gergaud@toulouse-inp.fr>
Date: Thu, 6 Oct 2022 11:08:44 +0200
Subject: [PATCH] TP non visibles

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-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<center>\n",
-    "<h1> Résolution de l'équation de la chaleur par discrétisation différences finies en temps et en espace</h1>\n",
-    "<h1> Année 2021-2022 - IENM2 </h1>\n",
-    "<h1> Nom:  </h1>\n",
-    "<h1> Prénom:  </h1>    \n",
-    "</center>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Objectifs "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<div style=\"background:MistyRose\">\n",
-    "<br>\n",
-    "Nous souhaitons résoudre le problème de la chaleur muni de conditions limites de type Dirichlet homogènes suivant:\n",
-    "<br>    \n",
-    "$$ (P)~ \\left \\{\n",
-    "        \\begin{array}{11}\n",
-    "            \\ \\displaystyle \\frac{\\partial u}{\\partial t}-\\Delta u = f \\, \\, \\, \\, \\, \\, (\\Omega \\times [0, T])\\\\\n",
-    "            \\ u(0,t)=0 \\, \\, \\, \\, \\, \\, (\\partial \\Omega) \\\\\n",
-    "            \\ u(1,t)=0 \\, \\, \\, \\, \\, \\, (\\partial \\Omega) \\\\\n",
-    "            \\ u(x,0)=u_{0}(x) \\, \\, \\, \\, \\, \\, (t=0)\n",
-    "         \\end{array} \n",
-    "         \\right. $$\n",
-    "<br> \n",
-    "sur un domaine monodimensionnel $\\Omega$ par une méthode de discrétisation de type différences finies.  Nous vous proposons de détailler pas à pas les différentes étapes en réalisant \n",
-    "systématiquement des tests unitaires afin de valider vos implantations.<br>  \n",
-    "\n",
-    "Les étapes proposées sont les suivantes:<br>\n",
-    "- Imposer une solution exacte $ u_{ex}(x,t)$ au problème $(P)$. \n",
-    "- En déduire une condition initiale $ u_{0}(x)$ au problème $(P)$. \n",
-    "- En déduire $f$ le terme source associé à $ u_{ex}(x,t)$. \n",
-    "- Résoudre numériquement $(P)$ en utilisant les schémas d'Euler explicite et implicite. \n",
-    "- Déterminer numériquement l'ordre de convergence des schémas de discrétisation en temps. \n",
-    "\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Import des librairies Python nécessaires"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "import scipy as sc\n",
-    "import scipy.sparse as sparse\n",
-    "import scipy.sparse.linalg\n",
-    "import numpy as np\n",
-    "import matplotlib.pyplot as plt\n",
-    "#import matplotlib.animation as animation\n",
-    "import math"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Définition des paramètres du problème (P)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {},
-   "outputs": [
-    {
-     "ename": "SyntaxError",
-     "evalue": "invalid syntax (<ipython-input-2-e30d73daff6c>, line 2)",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;36m  File \u001b[0;32m\"<ipython-input-2-e30d73daff6c>\"\u001b[0;36m, line \u001b[0;32m2\u001b[0m\n\u001b[0;31m    N_t =\u001b[0m\n\u001b[0m          ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
-     ]
-    }
-   ],
-   "source": [
-    "# Nombre de pas de discretisation en temps \n",
-    "N_t = \n",
-    "# Nombre de pas de discretisation en espace \n",
-    "M = \n",
-    "# Nombre de points de discretisation en espace\n",
-    "N = M + 1\n",
-    "# Temps final [s]\n",
-    "T = \n",
-    "# Pas de discretisation temporel sur [0 T]\n",
-    "delta_t = T/N_t\n",
-    "# Pas de discretisation spatial sur le domaine [0,1]\n",
-    "h = 1/(N-1)\n",
-    "# Nombre CFL \n",
-    "c = "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Définition de la fonction solution exacte choisie, du second membre de l'équation de la chaleur et déduction du schéma de construction associé à la solution exacte "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<div style=\"background:LightGrey\">\n",
-    "Indiquer ici le choix de la  solution exacte. En déduire le second membre de l'équation définissant le problème (P).\n",
-    "</div>    "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "def solution_exacte(x,t):\n",
-    "    \"\"\"\n",
-    "    Fonction representant la solution exacte du probleme de la chaleur. \n",
-    "    Cette fonction doit verifier les conditions initiales et limites sur \n",
-    "    la frontiere du domaine. \n",
-    "    Entrees:\n",
-    "    x: abscisse du point de maillage [float]\n",
-    "    t: temps discret considere [float]\n",
-    "    Sortie:\n",
-    "    Valeur de la solution exacte evaluee en x et au temps t [float]\n",
-    "    \"\"\"\n",
-    "    # A COMPLETER\n",
-    "\n",
-    "def second_membre(x, t):\n",
-    "    \"\"\"\n",
-    "    Fonction representant le second membre associe a la solution exacte du probleme \n",
-    "    de la chaleur.  \n",
-    "    Entrees:\n",
-    "    x: abscisse du point de maillage [float]\n",
-    "    t: temps discret considere [float]\n",
-    "    Sortie:\n",
-    "    Valeur du second membre evalue en x et au temps t [float]\n",
-    "    \"\"\"\n",
-    "    # A COMPLETER\n",
-    "\n",
-    "def schema_exact(N, h, c, T, delta_t, u0):\n",
-    "    \"\"\"\n",
-    "    Fonction representant le tableau bidimensionnel [espace, temps] correspondant a la \n",
-    "    solution en chaque point du maillage et a tout temps discret.\n",
-    "    Entrees:\n",
-    "    N: Nombre de points de discretisation en espace [int]\n",
-    "    h: Pas de maillage spatial [float]\n",
-    "    c: Nombre CFL [float]\n",
-    "    T: Temps final [float]\n",
-    "    delta_t: Pas de discretisation temporel [float]\n",
-    "    u0: condition initiale [array]\n",
-    "    Sortie:\n",
-    "    U_exacte: Tableau bidimensionnel de la solution evaluee en tout x et a tout temps t [float]\n",
-    "    \"\"\"\n",
-    "    # Initialisation de la solution \n",
-    "    \n",
-    "    # Prise en compte de la solution initiale\n",
-    "    \n",
-    "    # Deduction de la solution aux autres instants \n",
-    "            "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Définition de la condition initiale et affichage de la solution exacte pour différents instants "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "#\n",
-    "# Definition de la condition initiale comme un tableau monodimensionnel en espace\n",
-    "#\n",
-    "u0  = np.zeros(N, dtype=float)\n",
-    "# A COMPLETER pour avoir votre condition initiale\n",
-    "\n",
-    "#\n",
-    "# Deduction de la solution exacte par appel de la fonction schema_exact\n",
-    "#\n",
-    "U = schema_exact(N, h, c, T, delta_t, u0)\n",
-    "#\n",
-    "# Representation graphique a differents instants a choisir\n",
-    "#\n",
-    "x = np.linspace(0, 1, N)\n",
-    "plt.plot(x, U[:,0], label=\"à T=0\")\n",
-    "plt.plot(x, U[:, int(N_t/5)], label=\"à T/5\")\n",
-    "plt.plot(x, U[:, int(N_t/2)], label=\"à T/2\")\n",
-    "plt.plot(x, U[:,int(T/delta_t)-1], label=\"à T\" )\n",
-    "plt.legend()\n",
-    "plt.grid(True)\n",
-    "plt.title(\"Evolution de la température sur la barre pour T=\" + str(T))\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Construction du schéma explicite en temps"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "def schema_explicite(N, h, c, T, delta_t, U_explicite):\n",
-    "    \"\"\"\n",
-    "    Fonction representant le tableau bidimensionnel [espace, temps] correspondant a la \n",
-    "    solution en chaque point du maillage et a tout temps discret obtenue avec le schema \n",
-    "    explicite.\n",
-    "    Entrees:\n",
-    "    N: Nombre de points de discretisation en espace [int]\n",
-    "    h: Pas de maillage spatial [float]\n",
-    "    c: Nombre CFL [float]\n",
-    "    T: Temps final [float]\n",
-    "    delta_t: Pas de discretisation temporel [float]\n",
-    "    u0: condition initiale [array]\n",
-    "    Sortie:\n",
-    "    U_explicite: Tableau bidimensionnel de la solution du schema explicite evaluee en tout x et a tout temps t [float]\n",
-    "    \"\"\"\n",
-    "\n",
-    "    #\n",
-    "    # Création de la matrice A via une matrice creuse (interieur du domaine)\n",
-    "    #\n",
-    "    \n",
-    "    \n",
-    "    #\n",
-    "    # Application du schéma sur les points interieurs\n",
-    "    #\n",
-    "    "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Affichage de la solution discrète pour le schéma explicite pour différents instants "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [],
-   "source": [
-    "#\n",
-    "# La condition initiale est celle qui a été définie ci-dessus\n",
-    "\n",
-    "#\n",
-    "# Deduction de la solution du schema explicite en temps\n",
-    "#\n",
-    "U_explicite      = np.zeros((N, int(T/delta_t)), dtype=float)\n",
-    "U_explicite[:,0] = u0\n",
-    "Ue               = schema_explicite(N,h,c, T, delta_t, U_explicite)\n",
-    "#\n",
-    "# Representation graphique a differents instants a choisir\n",
-    "#\n",
-    "x = np.linspace(0, 1, N)\n",
-    "plt.plot(x, Ue[:,0], label=\"à T=0\")\n",
-    "plt.plot(x, Ue[:, int(N_t/5)], label=\"à T/5\")\n",
-    "plt.plot(x, Ue[:, int(N_t/2)], label=\"à T/2\")\n",
-    "plt.plot(x, Ue[:,int(T/delta_t)-1], label=\"à T\" )\n",
-    "plt.legend()\n",
-    "plt.grid(True)\n",
-    "plt.title(\"Evolution de la température sur la barre pour T=\" + str(T))\n",
-    "plt.show()\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Construction du  schéma implicite en temps"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "def schema_implicite(N,h,c, T, delta_t, U_implicite):\n",
-    "    \"\"\"\n",
-    "    Fonction representant le tableau bidimensionnel [espace, temps] correspondant a la \n",
-    "    solution en chaque point du maillage et a tout temps discret obtenue avec le schema \n",
-    "    explicite.\n",
-    "    Entrees:\n",
-    "    N: Nombre de points de discretisation en espace [int]\n",
-    "    h: Pas de maillage spatial [float]\n",
-    "    c: Nombre CFL [float]\n",
-    "    T: Temps final [float]\n",
-    "    delta_t: Pas de discretisation temporel [float]\n",
-    "    u0: condition initiale [array]\n",
-    "    Sortie:\n",
-    "    U_implicite: Tableau bidimensionnel de la solution du schema implicite evaluee en tout x et a tout temps t [float]\n",
-    "    \"\"\"\n",
-    "    #\n",
-    "    # Création de la matrice A via une matrice creuse\n",
-    "    #\n",
-    "    \n",
-    "    #\n",
-    "    # Application du schéma sur les points interieurs\n",
-    "    # \n",
-    "    "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Affichage de la solution discrète pour le schéma implicite pour différents instants "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "#\n",
-    "# La condition initiale est celle qui a été définie ci-dessus\n",
-    "\n",
-    "#\n",
-    "# Deduction de la solution du schema implicite en temps\n",
-    "#\n",
-    "U_implicite      = np.zeros((N, int(T/delta_t)), dtype=float)\n",
-    "U_implicite[:,0] = u0\n",
-    "Ui               = schema_implicite(N,h,c, T, delta_t, U_implicite)\n",
-    "#\n",
-    "# Representation graphique a differents instants a choisir\n",
-    "#\n",
-    "x = np.linspace(0, 1, N)\n",
-    "plt.plot(x, Ui[:,0], label=\"à T=0\")\n",
-    "plt.plot(x, Ui[:, int(N_t/5)], label=\"à T/5\")\n",
-    "plt.plot(x, Ui[:, int(N_t/2)], label=\"à T/2\")\n",
-    "plt.plot(x, Ui[:,int(T/delta_t)-1], label=\"à T\" )\n",
-    "plt.legend()\n",
-    "plt.grid(True)\n",
-    "plt.title(\"Evolution de la température sur la barre pour T=\" + str(T))\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Détermination de l'ordre de convergence des schémas de discrétisation en temps"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<div style=\"background:LightGrey\">\n",
-    "Indiquer ici comment vous calculez cet ordre de convergence. Quelle est la valeur attendue pour les schémas d'Euler explicite ou implicite ? Vérifier ce résultat graphiquement. \n",
-    "</div>"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "def erreur(U_exacte, U, h):\n",
-    "    \"\"\"\n",
-    "    Calcul de l'erreur discrete \n",
-    "    Entrees:\n",
-    "    U_exacte: solution exacte du probleme considere [float]\n",
-    "    U       : solution discrete pour un schema donne [float]\n",
-    "    h       : pas de discretisation spatial [float]\n",
-    "    Sortie: \n",
-    "    norme de l'erreur discrete [float]\n",
-    "    \"\"\"\n",
-    "    # A COMPLETER"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "def ordre_discretisation_temps(h, N, T):\n",
-    "    \"\"\"\n",
-    "    Calcul de l'ordre de discretisation en temps\n",
-    "    Entrées : \n",
-    "        h: pas de discretisation spatial  [float]\n",
-    "        N: nombre de points de discretisation en espace [int]\n",
-    "        T: temps final [float]\n",
-    "    Sortie : \n",
-    "        time_step: pas de temps en loi log [float, array]\n",
-    "        erreur  : erreur en loi log[float, array]\n",
-    "    \"\"\"\n",
-    "    time_step = np.zeros(10)\n",
-    "    error     = np.zeros(10)\n",
-    "    count     = 0\n",
-    "    \n",
-    "    for loop in range(20, 120, 10):\n",
-    "        delta = 1./loop\n",
-    "        c     = delta/(h**2)\n",
-    "        #\n",
-    "        # Initialisation pour la solution exacte\n",
-    "        #\n",
-    "        u0       = np.zeros(N, dtype=float)\n",
-    "        # A COMPLETER\n",
-    "        U_exacte = schema_exact(N, h, c, T, delta, u0)          \n",
-    "        #\n",
-    "        # Initialisation pour la solution numérique\n",
-    "        #\n",
-    "        U_implicite   = np.zeros((N, int(T/delta)), dtype=float)\n",
-    "        u0            = np.zeros(N, dtype=float)\n",
-    "        # A COMPLETER\n",
-    "        U_implicite[:,0] = u0\n",
-    "        #\n",
-    "        #\n",
-    "        #\n",
-    "        U_implicite      = schema_implicite(N, h, c, T, delta, U_implicite)\n",
-    "        error[count]     = math.log(erreur(U_exacte, U_implicite, h))\n",
-    "        time_step[count] = math.log(delta) \n",
-    "        count            += 1\n",
-    "        \n",
-    "    return time_step, error"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "#\n",
-    "# Construction des elements pour l'analyse du schema\n",
-    "#\n",
-    "time_step, error  = ordre_discretisation_temps(h, N, T)\n",
-    "\n",
-    "#\n",
-    "# Creation de la figure\n",
-    "#\n",
-    "theta = 1.0\n",
-    "fig = plt.figure()\n",
-    "plt.plot(time_step,error,marker='*',color='blue',label='Euler implicite (theta='\"%3.2f\"%theta+')')\n",
-    "plt.plot(time_step,1.*time_step,color='red',label='Ordre 1 en temps')\n",
-    "plt.plot(time_step,2.*time_step,color='green',label='Ordre 2 en temps')\n",
-    "plt.xlabel(\"Log(Delta t)\")\n",
-    "plt.ylabel(\"Log(Norme de l'erreur)\")\n",
-    "plt.title('Analyse de convergence: Equation de la chaleur')\n",
-    "plt.legend()\n",
-    "plt.grid(True)\n",
-    "plt.show()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<div style=\"background:LightGrey\">\n",
-    "Indiquer ici l'ordre de convergence obtenu effectivement. Obtenez-vous le comportement obtenu ? \n",
-    "</div>"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "#\n",
-    "# Affichage des donnees pour validation et inter-comparaison\n",
-    "#\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Schéma de discrétisation en temps du second ordre"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<div style=\"background:LightGrey\">\n",
-    "Résoudre (P) cette fois avec le schéma temporel d'ordre deux implicite Crank-Nicolson. Commenter votre code pour indiquer les changements effectués. Vérifier par une méthodologie identique l'ordre de convergence de ce schéma. \n",
-    "</div>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Bonus: Généralisation pour des conditions limites spatiales non homogènes"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "<div style=\"background:MistyRose\">\n",
-    "<br>\n",
-    "Nous souhaitons résoudre désormais le problème de la chaleur muni de conditions limites de \n",
-    "type Dirichlet non homogènes suivant:\n",
-    "<br>    \n",
-    "$$ (P_{nh})  ~  \\left \\{\n",
-    "        \\begin{array}{11}\n",
-    "            \\ \\displaystyle \\frac{\\partial u}{\\partial t}-\\Delta u = f \\, \\, \\, \\, \\, \\, (\\Omega \\times [0, T]),\\\\\n",
-    "            \\ u(0,t)= u_g{(t)} \\, \\, \\, \\, \\, \\, (\\partial \\Omega), \\\\\n",
-    "            \\ u(1,t)= u_d{(t)} \\, \\, \\, \\, \\, \\, (\\partial \\Omega), \\\\\n",
-    "            \\ u(x,0)= u_{0}(x) \\, \\, \\, \\, \\, \\, (t=0),\n",
-    "         \\end{array} \n",
-    "         \\right. $$\n",
-    "<br> \n",
-    "sur un domaine monodimensionnel $\\Omega$ par une méthode de discrétisation de type différences finies.<br>  \n",
-    "\n",
-    "Reprendre les étapes proposées précédemment:\n",
-    "\n",
-    "- Imposer une condition initiale et une solution exacte $ u_{ex}(x,t)$ au problème $(P_{nh})$. \n",
-    "- En déduire $f$ le terme source associé à $ u_{ex}$. \n",
-    "- Résoudre numériquement $(P_{nh})$ en utilisant un schéma de votre choix. \n"
-   ]
-  },
-  {
-   "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": "3.6.10"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
diff --git a/EDP/slide-intro-TP.pdf b/EDP/slide-intro-TP.pdf
deleted file mode 100644
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