{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "a3c831d8", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.integrate import solve_ivp\n", "\n", "#definition du système différentiel\n", "# y[0] = y\n", "# y[1] = y'\n", "# y\" = -y s'écrit y[0]' = y[1] et y[1]' = -y[0] \n", "\n", "def systeme(t, y):\n", " return [y[1], -y[0]]\n", "\n", "t_span = [0, 2*np.pi] # Intervalle de temps (une période)\n", "y0 = [0, 1] # Conditions initiales\n", "\n", "#résolution du système sur une période\n", "sol = solve_ivp(systeme, t_span, y0, t_eval=np.linspace(0, 2*np.pi, 100), method='RK45')\n", "t = sol.t\n", "y = sol.y\n", "#tracé des figures\n", "plt.subplot(121)\n", "plt.plot(t,y[0])\n", "plt.plot(t,y[1])\n", "plt.xlabel('t')\n", "plt.legend(['y[0]','y[1]'] )\n", "plt.subplot(122)\n", "plt.plot(y[0], y[1])\n", "plt.gca().set_aspect('equal', adjustable='box') # Définit les échelles des axes x et y égales\n", "plt.xlabel('y')\n", "plt.ylabel('dy/dt',)\n", "plt.title('Trajectoire de y')\n", "plt.grid(True)\n", "plt.subplots_adjust(wspace=0.6)# écarte un peu les deux figures\n", "plt.show()\n" ] }, { "cell_type": "markdown", "id": "365e5142", "metadata": {}, "source": [ "On obtient bien un cercle dans le plan des phases $(y,y')$.\n", "On a en effet $$y^2(t) + y'^2(t) \\equiv 1.$$\n", "En effet $$\\frac{d}{dt} (y^2(t) + y'^2(t)) = 2 y'(t)(y(t) + y\"(t)) =0. $$\n", "Donc $$y^2(t) + y'^2(t) = y^2(0) + y'^2(0)=1.$$\n", "Ici bien sûr $y(t) = \\sin t$ et $y'(t) = \\cos t$." ] }, { "cell_type": "code", "execution_count": 2, "id": "721217dd", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t_span = [0, 2000*np.pi] # Intervalle de temps (mille périodes)\n", "y0 = [0, 1] # Conditions initiales\n", "\n", "#résolution du système sur mille périodes\n", "sol = solve_ivp(systeme, t_span, y0, t_eval=np.linspace(0, 2000*np.pi, 200000), method='RK45')\n", "y = sol.y\n", "#tracé des figures\n", "plt.plot(y[0], y[1])\n", "plt.gca().set_aspect('equal', adjustable='box') # Définit les échelles des axes x et y égales\n", "plt.xlabel('y')\n", "plt.ylabel('dy/dt',)\n", "plt.title('Trajectoire de y')\n", "plt.grid(True)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "f0ff8947", "metadata": {}, "source": [ "On constate que le cercle s'épaissit. Les trajectoires ne se referment pas exactement, le schéma discret ne conserve pas l'intégrale première. Il perd peu à peu de l'énergie alors que le système différentiel continu n'en perd pas." ] }, { "cell_type": "code", "execution_count": 3, "id": "fc538bd9", "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t_span = [0, 2000*np.pi] # Intervalle de temps (une période)\n", "y0 = [0, 1] # Conditions initiales\n", "\n", "#résolution du système sur mille période\n", "sol = solve_ivp(systeme, t_span, y0, t_eval=np.linspace(0, 2000*np.pi, 200000), method='RK45', rtol=1e-5)\n", "y = sol.y\n", "#tracé des figures\n", "plt.plot(y[0], y[1])\n", "plt.gca().set_aspect('equal', adjustable='box') # Définit les échelles des axes x et y égales\n", "plt.xlabel('y')\n", "plt.ylabel('dy/dt',)\n", "plt.title('Trajectoire de y')\n", "plt.grid(True)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "6490b3f2", "metadata": {}, "source": [ "En augmentant la précision, le problème s'est atténué mais il n'a pas disparu. D'ailleurs si on refait la simulation sur 10 000 périodes, on retrouve l'épaississement du cercle." ] }, { "cell_type": "code", "execution_count": 4, "id": "79500b3c", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t_span = [0, 20000*np.pi] # Intervalle de temps (une période)\n", "y0 = [0, 1] # Conditions initiales\n", "\n", "#résolution du système sur mille période\n", "sol = solve_ivp(systeme, t_span, y0, t_eval=np.linspace(0, 20000*np.pi, 2000000), method='RK45', rtol=1e-5)\n", "y = sol.y\n", "#tracé des figures\n", "plt.plot(y[0], y[1])\n", "plt.gca().set_aspect('equal', adjustable='box') # Définit les échelles des axes x et y égales\n", "plt.xlabel('y')\n", "plt.ylabel('dy/dt',)\n", "plt.title('Trajectoire de y')\n", "plt.grid(True)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "dabd5e16", "metadata": {}, "source": [ "# Schéma implicite-explicite" ] }, { "cell_type": "code", "execution_count": 5, "id": "cd461486", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# duree de simulation\n", "T = 20000*np.pi\n", "# pas\n", "h = 1e-1\n", "# nombre de points de la subdivision\n", "N = int(T/h)\n", "q = np.zeros(N)\n", "p = np.zeros(N)\n", "# conditions initiales\n", "q[0] = 0\n", "p[0] = 1\n", "# schéma Euler symplectique\n", "# q_{n+1} = q_n + h p_{n+1}\n", "# p_{n+1} = p_n - h q_n\n", "for k in range(N-1):\n", " p[k+1] = p[k] - h*q[k]\n", " q[k+1] = q[k] + h*p[k+1]\n", "\n", "plt.figure()\n", "plt.plot(q, p, '-')\n", "plt.axis('equal')\n", "plt.xlabel('q')\n", "plt.ylabel('p')\n", "plt.title('schéma Euler implicite-explicite')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "8e1f40c7", "metadata": {}, "source": [ "Le problème est résolu, même avec un pas très grossier. Le schéma implicite-explicite conserve l'énergie, même si on simule sur dix mille périodes. C'est pourtant un schéma d'ordre 1 seulement comme le schéma d'Euler. L'ordre d'un schéma garantit sa précision mais pas la conservation des invariants du système. Les schémas qui respectent les invariants naturels du système sont dits ***symplectiques.***" ] }, { "cell_type": "code", "execution_count": null, "id": "1bfe11c2", "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 }