Ordinary-differential-equations.ipynb

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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "from bokeh.io import show, output_notebook\n",
    "from bokeh.plotting import figure\n",
    "from bokeh.layouts import row, column, gridplot\n",
    "\n",
    "output_notebook(hide_banner=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Méthode de Runge-Kutta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## La méthode RK2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Implémenter la méthode RK2. Tester le cas $f(t,y)=y$, pour $t_0=0$, $y_0=1$ et $T=1,$.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_rk2(f, t, y, h):\n",
    "    \"\"\"retourne l'approximation yn+1 de y au temps tn+1 = tn + h avec une méthode de Runge-Kutta d'ordre 2\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    f : fonction f(t,y) second membre de l'EDO\n",
    "    t : temps tn\n",
    "    y : valeur de yn\n",
    "    h : pas de temps\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    ynp1 : valeur de yn+1\n",
    "    \"\"\"\n",
    "    ynp1 = y + h*f( t+h/2, y + h/2*f(t,y) )\n",
    "    return ynp1\n",
    "\n",
    "\n",
    "def rk2(f, tini, tend, nt, yini):\n",
    "    \"\"\"retourne le vecteur contenant les valeurs des yn en appelant update_rk2\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    f : fonction f(t,y) second membre de l'EDO\n",
    "    tini : temps initial\n",
    "    tend : temps final\n",
    "    nt : dimension du vecteur de sortie\n",
    "    yini : valeur initiale y(tini)\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    y : vecteur les nt valeurs yn\n",
    "    \"\"\"\n",
    "\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    y = np.zeros(nt)\n",
    "    # au cas où la dimension de yini est supérieure à 1 (exercice 3)\n",
    "    if type(yini) == np.ndarray :\n",
    "        y = np.zeros( (nt,len(yini)) )\n",
    "    #----------------------------------\n",
    "    y[0] = yini\n",
    "    h = t[1]-t[0]\n",
    "    \n",
    "    for n in range(1,nt) :\n",
    "        y[n] = update_rk2(f,t[n-1],y[n-1],h)\n",
    "    \n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(t,y) :\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "BokehDeprecationWarning: 'legend' keyword is deprecated, use explicit 'legend_label', 'legend_field', or 'legend_group' keywords instead\n",
      "BokehDeprecationWarning: 'legend' keyword is deprecated, use explicit 'legend_label', 'legend_field', or 'legend_group' keywords instead\n"
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       "  var render_items = [{\"docid\":\"b9652788-d673-4c51-9a6b-03df82d8d997\",\"root_ids\":[\"1001\"],\"roots\":{\"1001\":\"dff37365-4104-49ba-ac73-12e780aec195\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
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      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1001"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "tini = 0\n",
    "tend = 10\n",
    "nt = 50\n",
    "yini = 1\n",
    "\n",
    "# Approximation numérique de la solution de l'EDO en utilisant la méthode de RK2\n",
    "y = rk2(f, tini, tend, nt, yini)\n",
    "t = np.linspace(tini, tend, nt)\n",
    "\n",
    "# La fonction exacte solution de l'EDO\n",
    "tsol = np.linspace(tini, tend, 200)\n",
    "ysol = np.exp(tsol)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(t,y, legend = \"y_RK2\")\n",
    "fig.line(tsol,ysol, color=\"orange\", legend = \"exp(t)\")\n",
    "fig.legend.location = \"top_left\"\n",
    "\n",
    "show(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "La solution générale de l'equation différentielle $y'=y$ est de la forme $y(t) = \\alpha e^{t}$, et la condition initiale $y(0)=1$ impose d'avoir $\\alpha=1$, donc $y(t) = e^{t}$  \n",
    "On remarque que l'approximation obtenue par la méthode de Runge-Kuta donne une courbe fiable de la solution de cette equation différentielle.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vérifier que l'erreur de convergence $\\max_{0\\leq n \\leq N } |y(t_n) - y_n| = O(h^2)$, c'est à dire que la méthode converge à l'ordre 2.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 304,
   "metadata": {},
   "outputs": [],
   "source": [
    "def max(x) :\n",
    "    m = x[0]\n",
    "    for i in range(len(x)) :\n",
    "        if x[i]>m : m=x[i]\n",
    "    return m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 305,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
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       "  <div class=\"bk-root\" id=\"a3568447-8dce-4c1d-ae65-bb38fc8d1987\" data-root-id=\"329621\"></div>\n"
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       "  var render_items = [{\"docid\":\"da9ffb0f-f684-4e7d-90c4-eafcedfe9bcd\",\"roots\":{\"329621\":\"a3568447-8dce-4c1d-ae65-bb38fc8d1987\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
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       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
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       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
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   "source": [
    "tini = 0\n",
    "tend = 1\n",
    "yini = 1\n",
    "\n",
    "# le tableau \"entiers\" contient les valeurs de nt, et E l'erreur de convergence associées à chaque valeur\n",
    "N = 200\n",
    "ntini =10\n",
    "pas = 10\n",
    "\n",
    "entiers = np.array([n for n in range(ntini, N, pas)])\n",
    "E = np.zeros(len(entiers))\n",
    "\n",
    "# Remplissage de E\n",
    "for i in range(len(entiers)) :\n",
    "    nt = entiers[i]\n",
    "    y = rk2(f, tini, tend, nt, yini)\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    ysol = np.exp(t)\n",
    "    err = abs(ysol-y)\n",
    "    E[i] = max(err)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "C1 = E[0]*entiers[0];  C2 = E[0]*entiers[0]**2;  C3 = E[0]*entiers[0]**3\n",
    "fig.line(entiers,C1/entiers, color=\"yellow\", legend=\"ordre 1\")\n",
    "fig.line(entiers,C2/entiers**2, color=\"orange\",legend=\"ordre 2\")\n",
    "fig.line(entiers,C3/entiers**3, color=\"pink\", legend=\"ordre 3\")\n",
    "fig.x(entiers,E, legend = \"max|y(t)-yn|_RK4\")\n",
    "fig.legend.location = \"bottom_left\"\n",
    "\n",
    "show(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "J'ai représenté sur la figure ci-dessus chacun des ordres 1,2 et 3, et j'ai marqué les points où j'ai calculé l'erreur de convergence. \n",
    "On remarque alors que la convergence est d'ordre 2.  \n",
    "**Remarque :** Les constantes C1,C2, et C3 que j'ai introduites dans mon programme servent uniquement à faire coincider toutes les courbes au premier point. Elles n'ont pas d'effet sur le résultat recherché car elles ne modifient pas les pentes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## La méthode RK4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Implémenter la méthode RK4 et vérifier numériquement sur le même exemple qu'à la question précédente.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 306,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_rk4(f, t, y, h):\n",
    "    \"\"\"retourne l'approximation yn+1 de y au temps tn+1 = tn + h avec une méthode de Runge-Kutta d'ordre 4\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    f : fonction f(t,y) second membre de l'EDO\n",
    "    t : temps tn\n",
    "    y : valeur de yn\n",
    "    h : pas de temps\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    ynp1 : valeur de yn+1\n",
    "    \"\"\"\n",
    "    \n",
    "    k1 = f(t,y)\n",
    "    k2 = f(t+h/2, y+h/2*k1)\n",
    "    k3 = f(t+h/2, y+h/2*k2)\n",
    "    k4 = f(t+h, y+h*k3)\n",
    "    ynp1 = y + h/6*(k1 + 2*k2 + 2*k3 + k4)\n",
    "    return ynp1\n",
    "\n",
    "\n",
    "def rk4(f, tini, tend, nt, yini):\n",
    "    \"\"\"retourne le vecteur contenant les valeurs des yn en appelant update_rk4\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    f : fonction f(t,y) second membre de l'EDO\n",
    "    tini : temps initial\n",
    "    tend : temps final\n",
    "    nt : dimension du vecteur de sortie\n",
    "    yini : valeur initiale y(tini)\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    y : vecteur les nt valeurs yn\n",
    "    \"\"\"\n",
    "    \n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    y = np.zeros(nt)\n",
    "    # au cas où la dimension de yini est supérieure à 1 (exercice 3)\n",
    "    if type(yini) == np.ndarray :\n",
    "        y = np.zeros( (nt,len(yini)) )\n",
    "    #----------------------------------\n",
    "    y[0] = yini\n",
    "    h = t[1]-t[0]\n",
    "    \n",
    "    for n in range(1,nt) :\n",
    "        y[n] = update_rk4(f,t[n-1],y[n-1],h)\n",
    "    \n",
    "    return y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Vérifier que cette méthode est d'ordre 4.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 307,
   "metadata": {},
   "outputs": [
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       "\n",
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       "\n",
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"type\":\"Line\"},{\"attributes\":{\"label\":{\"value\":\"max|y(t)-yn|_RK4\"},\"renderers\":[{\"id\":\"333556\",\"type\":\"GlyphRenderer\"}]},\"id\":\"333571\",\"type\":\"LegendItem\"},{\"attributes\":{\"callback\":null},\"id\":\"333478\",\"type\":\"DataRange1d\"},{\"attributes\":{\"label\":{\"value\":\"ordre 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Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"f533392e-cefa-46d2-92bd-7b3437f67c70\",\"roots\":{\"333475\":\"3138198b-a853-46cc-ac6c-f24bcde1256c\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "333475"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "tini = 0\n",
    "tend = 1\n",
    "yini = 1\n",
    "\n",
    "# le tableau \"entiers\" contient les valeurs de nt, et E l'erreur de convergence associées à chaque valeur\n",
    "N = 100\n",
    "ntini =10\n",
    "pas = 10\n",
    "\n",
    "entiers = np.array([n for n in range(ntini, N, pas)])\n",
    "E = np.zeros(len(entiers))\n",
    "\n",
    "# Remplissage de E\n",
    "for i in range(len(entiers)) :\n",
    "    nt = entiers[i]\n",
    "    y = rk4(f, tini, tend, nt, yini)\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    ysol = np.exp(t)\n",
    "    err = abs(ysol-y)\n",
    "    E[i] = max(err)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "C1 = E[0]*entiers[0]**3;  C2 = E[0]*entiers[0]**4;  C3 = E[0]*entiers[0]**5\n",
    "fig.line(entiers,C1/entiers**3, color=\"yellow\", legend=\"ordre 3\")\n",
    "fig.line(entiers,C2/entiers**4, color=\"orange\",legend=\"ordre 4\")\n",
    "fig.line(entiers,C3/entiers**5, color=\"pink\", legend=\"ordre 5\")\n",
    "fig.x(entiers,E, legend = \"max|y(t)-yn|_RK4\")\n",
    "fig.legend.location = \"bottom_left\"\n",
    "\n",
    "show(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "On remarque que l'ordre de convergence est 4.  \n",
    "**Remarque :** Les points obtenus sont (légèrement) en dessous de la droite correspondant à l'ordre 4, et l'ordre étant l'entier le plus grand entier vérifiant cette propriété, on déduit que l'ordre est bien 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tableau de Butcher "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Bonus : Implémenter une méthode de Runge-Kutta générale explicite utilisant le tableau de Butcher. On implementera une fonction update_rk qui retourne $y_{n+1}$ étant données $f,t_n,y_n,h$ et les tableaux $A,B,C$ stockant respectivement les coefficients $(a_{ij})_{1 \\leq i,j \\leq p}$, $(b_j)_{j=1,...,p}$ et $(c_i)_{i=1,...,p}$ décrits dans le sujet de la PC.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 308,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_rk(f, t, y, h, A, B, C):\n",
    "    \"\"\"retourne l'approximation yn+1 de y au temps tn+1 = tn + h avec une méthode de Runge-Kutta \n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    f : fonction f(t,y) second membre de l'EDO\n",
    "    t : temps tn\n",
    "    y : valeur de yn\n",
    "    h : pas de temps\n",
    "    A : matrice A du tableau de Butcher (incluant les zéros)\n",
    "    B : vecteur B du tableau de Butcher \n",
    "    C : vecteur C du tableau de Butcher (incluant c1 = 0)\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    ynp1 : valeur de yn+1\n",
    "    \"\"\"\n",
    "    p = len(A)\n",
    "    k = np.zeros(p)\n",
    "    \n",
    "    # Calcul de k_i\n",
    "    for i in range(1,p+1) :\n",
    "        s = 0\n",
    "        for j in range(1,i) :\n",
    "            s += A[i-1][j-1]*k[j-1]\n",
    "        k[i-1] = f(t+C[i-1]*h, y + h*s)\n",
    "    \n",
    "    s = 0\n",
    "    for j in range(p) :\n",
    "        s += B[j]*k[j]\n",
    "    ynp1 = y + h*s\n",
    "    \n",
    "    return ynp1\n",
    "\n",
    "\n",
    "def rk(f, tini, tend, nt, yini, A, B, C):\n",
    "    \"\"\"retourne le vecteur contenant les valeurs des yn en appelant update_rk\n",
    "       \n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    f : fonction f(t,y) second membre de l'EDO\n",
    "    tini : temps initial\n",
    "    tend : temps final\n",
    "    nt : dimension du vecteur de sortie\n",
    "    yini : valeur initiale y(tini)\n",
    "    A : matrice A du tableau de Butcher (incluant les zéros)\n",
    "    B : vecteur B du tableau de Butcher \n",
    "    C : vecteur C du tableau de Butcher (incluant c1 = 0)\n",
    "\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    y : vecteur les nt valeurs yn\n",
    "    \"\"\"\n",
    "\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    y = np.zeros(nt)\n",
    "    y[0] = yini\n",
    "    h = t[1]-t[0]\n",
    "    \n",
    "    for n in range(1,nt) :\n",
    "        y[n] = update_rk(f,t[n-1],y[n-1],h,A,B,C)\n",
    "    \n",
    "    return y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Bonus : Tester avec les tableaux de Butcher correspondant aux méthodes rk2 et rk4 déjà implémentés. Vérifier que vous obtenez les mêmes résultats sur le cas $f(t,y)=y$.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 309,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"44d9de1b-2d77-41ac-a9f2-ae2a4789443d\" data-root-id=\"337502\"></div>\n"
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     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
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       "  var render_items = [{\"docid\":\"2b016034-2be6-4b03-b5f1-9ff9ca704a28\",\"roots\":{\"337502\":\"44d9de1b-2d77-41ac-a9f2-ae2a4789443d\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
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       "  if (root.Bokeh !== undefined) {\n",
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       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
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       "        embed_document(root);\n",
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       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
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       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
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      "application/vnd.bokehjs_exec.v0+json": ""
     },
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       "id": "337502"
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     "output_type": "display_data"
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   "source": [
    "### RK2 en utilisant la fonction rk\n",
    "\n",
    "tini = 0\n",
    "tend = 1\n",
    "yini = 1\n",
    "nt = 10\n",
    "\n",
    "A = np.array([ [0,0],[1/2,0] ])\n",
    "B = np.array([0,1])\n",
    "C = np.array([0,1/2])\n",
    "\n",
    "\n",
    "\n",
    "# Approximation numérique de la solution de l'EDO en utilisant la méthode de RK2\n",
    "y = rk(f, tini, tend, nt, yini, A, B, C)\n",
    "t = np.linspace(tini, tend, nt)\n",
    "\n",
    "# La fonction exacte solution de l'EDO\n",
    "tsol = np.linspace(tini, tend, 200)\n",
    "ysol = np.exp(tsol)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(t,y, legend = \"y_RK2\")\n",
    "fig.line(tsol,ysol, color=\"orange\", legend = \"exp(t)\")\n",
    "fig.legend.location = \"top_left\"\n",
    "\n",
    "\n",
    "################## Erreur\n",
    "# le tableau \"entiers\" contient les valeurs de nt, et E l'erreur de convergence associées à chaque valeur\n",
    "N = 200\n",
    "ntini =10\n",
    "pas = 10\n",
    "\n",
    "entiers = np.array([n for n in range(ntini, N, pas)])\n",
    "E = np.zeros(len(entiers))\n",
    "\n",
    "# Remplissage de E\n",
    "for i in range(len(entiers)) :\n",
    "    nt = entiers[i]\n",
    "    y = rk(f, tini, tend, nt, yini, A, B, C)\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    ysol = np.exp(t)\n",
    "    err = abs(ysol-y)\n",
    "    E[i] = max(err)\n",
    "\n",
    "# Figure\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "C1 = E[0]*entiers[0];  C2 = E[0]*entiers[0]**2;  C3 = E[0]*entiers[0]**3\n",
    "fig2.line(entiers,C1/entiers, color=\"yellow\", legend=\"ordre 1\")\n",
    "fig2.line(entiers,C2/entiers**2, color=\"orange\",legend=\"ordre 2\")\n",
    "fig2.line(entiers,C3/entiers**3, color=\"pink\", legend=\"ordre 3\")\n",
    "fig2.x(entiers,E, legend = \"max|y(t)-yn|_RK2\")\n",
    "fig2.legend.location = \"bottom_left\"\n",
    "\n",
    "\n",
    "show(row(fig,fig2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 310,
   "metadata": {},
   "outputs": [
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       "  var render_items = [{\"docid\":\"05fd4ef3-2529-44f5-9808-620171c0b3f0\",\"roots\":{\"341525\":\"1c572084-d7a4-4b45-9023-719c4eff2d93\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "341525"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "### RK2 en utilisant la fonction rk\n",
    "\n",
    "tini = 0\n",
    "tend = 1\n",
    "yini = 1\n",
    "nt = 20\n",
    "\n",
    "A = np.array([ [0,0,0,0],[1/2,0,0,0],[0,1/2,0,0],[0,0,1,0] ])\n",
    "B = np.array([1/6, 1/3, 1/3, 1/6])\n",
    "C = np.array([0,1/2,1/2,1])\n",
    "\n",
    "\n",
    "# Approximation numérique de la solution de l'EDO en utilisant la méthode de RK4\n",
    "y = rk(f, tini, tend, nt, yini, A, B, C)\n",
    "t = np.linspace(tini, tend, nt)\n",
    "\n",
    "# La fonction exacte solution de l'EDO\n",
    "tsol = np.linspace(tini, tend, 200)\n",
    "ysol = np.exp(tsol)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(t,y, legend = \"y_RK2\")\n",
    "fig.line(tsol,ysol, color=\"orange\", legend = \"exp(t)\")\n",
    "fig.legend.location = \"top_left\"\n",
    "\n",
    "\n",
    "################## Erreur\n",
    "# le tableau \"entiers\" contient les valeurs de nt, et E l'erreur de convergence associées à chaque valeur\n",
    "N = 100\n",
    "ntini =10\n",
    "pas = 10\n",
    "\n",
    "entiers = np.array([n for n in range(ntini, N, pas)])\n",
    "E = np.zeros(len(entiers))\n",
    "\n",
    "# Remplissage de E\n",
    "for i in range(len(entiers)) :\n",
    "    nt = entiers[i]\n",
    "    y = rk(f, tini, tend, nt, yini, A, B, C)\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    ysol = np.exp(t)\n",
    "    err = abs(ysol-y)\n",
    "    E[i] = max(err)\n",
    "\n",
    "# Figure\n",
    "fig2 = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "C1 = E[0]*entiers[0]**3;  C2 = E[0]*entiers[0]**4;  C3 = E[0]*entiers[0]**5\n",
    "fig2.line(entiers,C1/entiers**3, color=\"yellow\", legend=\"ordre 3\")\n",
    "fig2.line(entiers,C2/entiers**4, color=\"orange\",legend=\"ordre 4\")\n",
    "fig2.line(entiers,C3/entiers**5, color=\"pink\", legend=\"ordre 5\")\n",
    "fig2.x(entiers,E, legend = \"max|y(t)-yn|_RK4\")\n",
    "fig2.legend.location = \"bottom_left\"\n",
    "\n",
    "show(row(fig,fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "En définissant correctement A,B et C, on obtient les mêmes résultats que précédemment."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Un système conservatif"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On considère l'ocillateur harmonique :\n",
    "\\begin{equation}\n",
    "\\left\\{\n",
    "\\begin{aligned}\n",
    "q'&=p\\\\\n",
    "p'&=-q.\n",
    "\\end{aligned}\n",
    "\\right.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Schéma d'Euler explicite et implicite\n",
    "\n",
    "**Implémenter les schémas d'Euler explicite et implicite pour le système précédent.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Commencons par implémenter la solution analytique du système 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 311,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sol2(t, qini, pini) :\n",
    "    tini = t[0]\n",
    "    a = np.cos(tini)*qini - np.sin(tini)*pini\n",
    "    b = np.sin(tini)*qini + np.cos(tini)*pini\n",
    "    \n",
    "    Cos = np.cos(t)\n",
    "    Sin = np.sin(t)\n",
    "    \n",
    "    q = a*Cos + b*Sin\n",
    "    p = -a*Sin + b*Cos\n",
    "    \n",
    "    return q, p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 312,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "La fonction euler_explicite/implicite retourne le couple (qnp1, pnp1) à l'instant t_{n+1}\n",
    "Paramètres :\n",
    "_ q = qn\n",
    "_ p = pn\n",
    "_ h = pas de temps\n",
    "\"\"\"\n",
    "# Euler explicite\n",
    "def update_Eexplicite(q, p, h):\n",
    "    qnp1 = q + h*p\n",
    "    pnp1 = p - h*q\n",
    "    return qnp1, pnp1\n",
    "\n",
    "def euler_explicite(pini, qini, yini, nt, h) :\n",
    "    qe = np.zeros(nt)\n",
    "    pe = np.zeros(nt)\n",
    "    qe[0], pe[0] = qini, pini\n",
    "    for n in range(1,nt) :\n",
    "        qe[n], pe[n] = update_Eexplicite(qe[n-1], pe[n-1], h)\n",
    "    return qe, pe\n",
    "    \n",
    "\n",
    "# Euler implicite\n",
    "def update_Eimplicite(p, q, h):\n",
    "    qnp1 = (-h*q + p)/(1+h*h)\n",
    "    pnp1 = (q + h*p)/(1+h*h)\n",
    "    return qnp1, pnp1\n",
    "\n",
    "def euler_implicite(pini, qini, yini, nt, h) :\n",
    "    qi = np.zeros(nt)\n",
    "    pi = np.zeros(nt)\n",
    "    qi[0], pi[0] = qini, pini\n",
    "    for n in range(1,nt) :\n",
    "        qi[n], pi[n] = update_Eimplicite(qi[n-1], pi[n-1], h)\n",
    "    return qi, pi\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Représenter les solutions $q_n$ et $p_n$ obtenues, ainsi que l'hamiltonien $\\mathcal{H}(q_n,p_n)$ associé. On pourra tracer $q$ et $p$ en fonction du temps, mais aussi tracer la solution dans le plan de phase $(q,p)$. Commenter les résultats obtenus.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 315,
   "metadata": {},
   "outputs": [
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\",\"dtype\":\"float64\",\"shape\":[1000]},\"y\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"shape\":[1000]},\"y\":{\"__ndarray__\":\"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\",\"dtype\":\"float64\",\"shape\":[1000]}},\"selected\":{\"id\":\"354365\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"354364\",\"type\":\"UnionRenderers\"}},\"id\":\"354306\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"354307\",\"type\":\"Line\"},{\"attributes\":{\"children\":[[{\"id\":\"354096\",\"subtype\":\"Figure\",\"type\":\"Plot\"},0,0],[{\"id\":\"354162\",\"subtype\":\"Figure\",\"type\":\"Plot\"},0,1],[{\"id\":\"354208\",\"subtype\":\"Figure\",\"type\":\"Plot\"},1,0],[{\"id\":\"354274\",\"subtype\":\"Figure\",\"type\":\"Plot\"},1,1]]},\"id\":\"354368\",\"type\":\"GridBox\"},{\"attributes\":{},\"id\":\"354213\",\"type\":\"LinearScale\"},{\"attributes\":{\"line_color\":\"orange\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"354143\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"354293\",\"type\":\"PanTool\"},{\"attributes\":{},\"id\":\"354215\",\"type\":\"LinearScale\"},{\"attributes\":{\"label\":{\"value\":\"euler_implicite\"},\"renderers\":[{\"id\":\"354243\",\"type\":\"GlyphRenderer\"}]},\"id\":\"354253\",\"type\":\"LegendItem\"},{\"attributes\":{\"items\":[{\"id\":\"354253\",\"type\":\"LegendItem\"},{\"id\":\"354268\",\"type\":\"LegendItem\"}],\"location\":\"top_left\"},\"id\":\"354252\",\"type\":\"Legend\"},{\"attributes\":{\"ticker\":{\"id\":\"354218\",\"type\":\"BasicTicker\"}},\"id\":\"354221\",\"type\":\"Grid\"},{\"attributes\":{},\"id\":\"354218\",\"type\":\"BasicTicker\"},{\"attributes\":{},\"id\":\"354294\",\"type\":\"WheelZoomTool\"},{\"attributes\":{\"formatter\":{\"id\":\"354250\",\"type\":\"BasicTickFormatter\"},\"ticker\":{\"id\":\"354223\",\"type\":\"BasicTicker\"}},\"id\":\"354222\",\"type\":\"LinearAxis\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"354144\",\"type\":\"Line\"},{\"attributes\":{\"overlay\":{\"id\":\"354317\",\"type\":\"BoxAnnotation\"}},\"id\":\"354295\",\"type\":\"BoxZoomTool\"},{\"attributes\":{\"dimension\":1,\"ticker\":{\"id\":\"354223\",\"type\":\"BasicTicker\"}},\"id\":\"354226\",\"type\":\"Grid\"},{\"attributes\":{\"data_source\":{\"id\":\"354142\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"354143\",\"type\":\"Line\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"354144\",\"type\":\"Line\"},\"selection_glyph\":null,\"view\":{\"id\":\"354146\",\"type\":\"CDSView\"}},\"id\":\"354145\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"active_drag\":\"auto\",\"active_inspect\":\"auto\",\"active_multi\":null,\"active_scroll\":\"auto\",\"active_tap\":\"auto\",\"tools\":[{\"id\":\"354293\",\"type\":\"PanTool\"},{\"id\":\"354294\",\"type\":\"WheelZoomTool\"},{\"id\":\"354295\",\"type\":\"BoxZoomTool\"},{\"id\":\"354296\",\"type\":\"SaveTool\"},{\"id\":\"354297\",\"type\":\"ResetTool\"},{\"id\":\"354298\",\"type\":\"HelpTool\"}]},\"id\":\"354299\",\"type\":\"Toolbar\"},{\"attributes\":{},\"id\":\"354296\",\"type\":\"SaveTool\"},{\"attributes\":{\"source\":{\"id\":\"354142\",\"type\":\"ColumnDataSource\"}},\"id\":\"354146\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"354297\",\"type\":\"ResetTool\"},{\"attributes\":{\"data_source\":{\"id\":\"354306\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"354307\",\"type\":\"Line\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"354308\",\"type\":\"Line\"},\"selection_glyph\":null,\"view\":{\"id\":\"354310\",\"type\":\"CDSView\"}},\"id\":\"354309\",\"type\":\"GlyphRenderer\"},{\"attributes\":{},\"id\":\"354223\",\"type\":\"BasicTicker\"},{\"attributes\":{\"line_alpha\":0.1,\"line_color\":\"#1f77b4\",\"x\":{\"field\":\"x\"},\"y\":{\"field\":\"y\"}},\"id\":\"354256\",\"type\":\"Line\"},{\"attributes\":{},\"id\":\"354298\",\"type\":\"HelpTool\"}],\"root_ids\":[\"354371\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"9b6289a2-c89e-47e1-b1bd-49f8b02c6ef9\",\"roots\":{\"354371\":\"801cdee9-bc7b-44fe-ae4f-5526ddc76337\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "354371"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "tini = 0\n",
    "tend = 40\n",
    "nt = 1000\n",
    "qini = 1\n",
    "pini = 0\n",
    "\n",
    "t = np.linspace(tini, tend, nt)\n",
    "h = t[1]-t[0]\n",
    "qsol, psol = sol2(t, qini, pini)\n",
    "\n",
    "# Euler explicite\n",
    "qe, pe = euler_explicite(pini, qini, yini, nt, h)\n",
    "He = (qe*qe+pe*pe)/2\n",
    "\n",
    "\n",
    "# Euler implicite\n",
    "qi, pi = euler_implicite(pini, qini, yini, nt, h)\n",
    "Hi = (qi*qi+pi*pi)/2\n",
    "\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(qe, pe, legend = \"euler_explicite\")\n",
    "fig.line(qsol, psol, color = \"orange\", legend = \"solution analytique\")\n",
    "fig.x(qe[0],pe[0], color = \"red\", size = 10)\n",
    "fig.legend.location = \"top_left\"\n",
    "fig2 = figure(width=490, height=300)\n",
    "fig2.line(t, He, legend =\"Energie_explicite\")\n",
    "fig2.legend.location = \"top_left\"\n",
    "\n",
    "figb = figure(width=490, height=300)\n",
    "figb.line(qi, pi, legend = \"euler_implicite\")\n",
    "figb.line(qsol, psol, color = \"orange\", legend = \"solution analytique\")\n",
    "figb.x(qi[0],pi[0], color = \"red\", size = 10)\n",
    "figb.legend.location = \"top_left\"\n",
    "figb2 = figure(width=490, height=300)\n",
    "figb2.line(t, Hi, legend =\"Energie_implicite\")\n",
    "\n",
    "\n",
    "grid = gridplot([[fig, fig2], [figb, figb2]])\n",
    "show(grid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**\n",
    "Théoriqement, l'énergie $H(q,p)$ est conservée, elle ne varie pas avec le temps, et par conséquent $\\forall t, (p(t)^2+q(t)^2)/2 = H_0$ où $H_0$ est la valeur initiale de l'énergie. Ceci montre que dans le plan de phase, l'ensemble des points $(q(t),p(t))$ forment une ellipse.  \n",
    "Or, en appliquant le shéma d'Euler implicite :\n",
    "$$q_{n+1} = q_n + hp_n$$ $$p_{n+1} = p_n - hq_n$$ Et on peut montrer par récurrence que $$H(q_n,p_n) = H_0(1+h)^n$$\n",
    "L'energie obtenue diverge avec le temps, c'est bien ce qu'on remarque sur la figure obtenue. Et on remarque aussi que $(q_n,p_n)$ est poussé vers l'extérieur au cours du temps.  \n",
    "Par contre, pour le shéma d'Euler implicite :\n",
    "$$q_{n+1} = (q_n + hp_n)/(1+h^2)$$ $$p_{n+1} = (p_n - hq_n)/(1+h^2)$$ Et on peut montrer par récurrence que $$H(q_n,p_n) = H_0/(1+h)^n$$\n",
    "L'energie cette fois tend vers 0, ce qui correspond à une atténuation de $q$, et $(q_n,p_n)$ est attiré au centre au cours du temps.  \n",
    "\n",
    "**Conclusion :** Aucun des deux shémas ne donne une approximation exacte de la solution de l'equation différentielle, toutefois, en augmentant assez la taille de la subdivision par rapport à la longueur de $[t_{ini},t_{end}]$, on peut obtenir une approximation aussi bonne que l'on souhaite. (par exemple pour tini = 0, tend = 10 et n = 5000, l'erreur est majorée par 0.1)  \n",
    "La solution recherchée est périodique de période $2\\pi$, on peut donc se contenter de prendre par exemple tini=0, tend=6.3 (6.3>2$\\pi$)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Schéma d'ordre élevé"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Appliquer le schéma RK2 (décrit dans l'exercice précédent) au système de l'ocillateur harmonique. Comparer les résultats $p_n$, $q_n$ et $\\mathcal{H}(q_n,p_n)$ avec ceux obtenus à la question précédente. Commenter les résultats obtenus.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 317,
   "metadata": {},
   "outputs": [
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id\":\"362869\",\"type\":\"LegendItem\"},{\"attributes\":{},\"id\":\"362831\",\"type\":\"LinearScale\"},{\"attributes\":{\"data_source\":{\"id\":\"362819\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"362820\",\"type\":\"X\"},\"hover_glyph\":null,\"muted_glyph\":null,\"nonselection_glyph\":{\"id\":\"362821\",\"type\":\"X\"},\"selection_glyph\":null,\"view\":{\"id\":\"362823\",\"type\":\"CDSView\"}},\"id\":\"362822\",\"type\":\"GlyphRenderer\"}],\"root_ids\":[\"362870\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"d737e1d1-d4c6-4694-a27d-ef3a98dd1348\",\"roots\":{\"362870\":\"5249cebc-d248-4127-bc83-f4a6ad32822e\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "362870"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "tini = 0\n",
    "tend = 10\n",
    "nt = 100\n",
    "qini = 1\n",
    "pini = 0\n",
    "\n",
    "yini = np.array([qini, pini])\n",
    "def f(t,y) :\n",
    "    return np.array([y[1], -y[0]])\n",
    "\n",
    "t = np.linspace(tini, tend, nt)\n",
    "\n",
    "# Choisir RK2 ou RK4 -------------------------------------------\n",
    "y = rk2(f, tini, tend, nt, yini)\n",
    "#y = rk4(f, tini, tend, nt, yini)\n",
    "#---------------------------------------------------------------\n",
    "\n",
    "q = np.array([y[i][0] for i in range(nt)])\n",
    "p = np.array([y[i][1] for i in range(nt)])\n",
    "H = (p*p + q*q)/2\n",
    "\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(q, p, legend = \"(q,p)\")\n",
    "fig.x(q[0],p[0], color = \"red\", size = 10)\n",
    "fig.legend.location = \"top_left\"\n",
    "fig2 = figure(width=490, height=300)\n",
    "fig2.line(t, H, legend =\"Energie\")\n",
    "\n",
    "show(row(fig,fig2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "L'equation différentielle vérifiée par $q$ peut se ramener à $y' = (y[1], -y[0]) = f(t,y) $ avec $y = (q,p)$.  \n",
    "J'ai adapté la fonction rk2 définie dans l'exercice 2 pour qu'elle puisse agir également sur des vecteurs.\n",
    "On remarque que la méthode de Runge-Kutta d'ordre 2 donne une meilleure approximation de la solution de l'EDP, car pour un rapport np/(tend-tini) = 10, on obtient une energie qui ne varie que très peu, et une solution très acceptable.  \n",
    "J'ai laissé dans mon code aussi la possibilité d'utiliser la méthode de RK-4, qui donne approximation encore meilleure que RK-2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Schéma symplectique\n",
    "\n",
    "Considérez le schéma suivant :\n",
    "\n",
    "\\begin{equation*}\n",
    "  \\label{schema_symp}\n",
    "  y_{n+1} = y_n + h f\\left(t_n+\\frac{h}{2},\\ \\frac{y_n+y_{n+1}}{2}\\right).\n",
    "\\end{equation*}\n",
    "\n",
    "**Implémenter ce schéma pour le système  l'ocillateur harmonique.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 318,
   "metadata": {},
   "outputs": [],
   "source": [
    "def update_pt_milieu_symplectique(q, p, h):\n",
    "    qnp1 = ( (1-h*h/4)*q + h*p )/(1+h*h/4)\n",
    "    pnp1 = ( (1-h*h/4)*p - h*q )/(1+h*h/4)\n",
    "    return qnp1, pnp1\n",
    "\n",
    "def pt_milieu_symplectique(pini, qini, yini, nt, h):\n",
    "    q = np.zeros(nt)\n",
    "    p = np.zeros(nt)\n",
    "    q[0], p[0] = qini, pini\n",
    "    for n in range(1,nt) :\n",
    "        q[n], p[n] = update_pt_milieu_symplectique(q[n-1], p[n-1], h)\n",
    "    return q, p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Observer numériquement son ordre, ainsi que l'évolution de l'hamiltonien, et commenter les résultats obtenus.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 319,
   "metadata": {},
   "outputs": [
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Ticker\"}},\"id\":\"366976\",\"type\":\"Grid\"},{\"attributes\":{},\"id\":\"366930\",\"type\":\"ResetTool\"},{\"attributes\":{},\"id\":\"366922\",\"type\":\"BasicTicker\"},{\"attributes\":{\"source\":{\"id\":\"366939\",\"type\":\"ColumnDataSource\"}},\"id\":\"366943\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"370935\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"366912\",\"type\":\"LinearScale\"},{\"attributes\":{\"callback\":null},\"id\":\"366959\",\"type\":\"DataRange1d\"}],\"root_ids\":[\"367004\"]},\"title\":\"Bokeh Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"e59a8f54-74e6-4ba1-83eb-79ce8ff3a1e7\",\"roots\":{\"367004\":\"6789ea09-b8ec-4b92-9c6d-e8ce33d6407d\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "367004"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "tini = 0\n",
    "tend = 10\n",
    "nt = 100\n",
    "qini = 1\n",
    "pini = 0\n",
    "\n",
    "t = np.linspace(tini, tend, nt)\n",
    "h = t[1]-t[0]\n",
    "\n",
    "# Euler explicite\n",
    "q, p = pt_milieu_symplectique(pini, qini, yini, nt, h)\n",
    "H = (q*q+p*p)/2\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300)\n",
    "fig.line(q, p, legend = \"euler_explicite\")\n",
    "fig.x(q[0],p[0], color = \"red\", size = 10)\n",
    "fig.legend.location = \"top_left\"\n",
    "fig2 = figure(width=490, height=300)\n",
    "fig2.line(t, H, legend =\"Energie_explicite\")\n",
    "fig2.legend.location = \"top_left\"\n",
    "\n",
    "show(row(fig,fig2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 299,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"911f1232-81ce-42c4-b12a-f50f8015cacf\" data-root-id=\"322014\"></div>\n"
      ]
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   "source": [
    "# le tableau \"entiers\" contient les valeurs de nt, et E l'erreur de convergence associées à chaque valeur\n",
    "N = 100\n",
    "ntini =10\n",
    "pas = 10\n",
    "entiers = np.array([n for n in range(ntini, N, pas)])\n",
    "\n",
    "# Erreur Euler_explicite --------------------------------------------\n",
    "E = np.zeros(len(entiers))\n",
    "# Remplissage de E\n",
    "for i in range(len(entiers)) :\n",
    "    nt = entiers[i]\n",
    "    t = np.linspace(tini, tend, nt)\n",
    "    h = t[1] - t[0]\n",
    "    q, p = pt_milieu_symplectique(pini, qini, yini, nt, h)\n",
    "    qsol, psol = sol2(t, qini, pini)\n",
    "    err = abs(q-qsol)\n",
    "    E[i] = max(err)\n",
    "\n",
    "# Figure\n",
    "fig = figure(width=490, height=300, x_axis_type=\"log\", y_axis_type=\"log\")\n",
    "C1 = E[0]*entiers[0];  C2 = E[0]*entiers[0]**2;  C3 = E[0]*entiers[0]**3\n",
    "fig.line(entiers,C1/entiers, color=\"yellow\", legend=\"ordre 1\")\n",
    "fig.line(entiers,C2/entiers**2, color=\"orange\",legend=\"ordre 2\")\n",
    "fig.line(entiers,C3/entiers**3, color=\"pink\", legend=\"ordre 3\")\n",
    "fig.x(entiers,E, legend = \"erreur pour euler_explicite\")\n",
    "fig.legend.location = \"bottom_left\"\n",
    "\n",
    "\n",
    "show(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Commentaire :**  \n",
    "La méthode du pont de milieu simplectique est donc d'ordre 2.  "
   ]
  }
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