{ "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.display import display, Latex\n", "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" ] } ], "source": [ "import sys #only needed to determine Python version number\n", "print('Python version ' + sys.version)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# M62_CM4 : Construction de schémas multipas" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Considérons le problème de Cauchy\n", "\n", "
\n", "\n", "trouver une fonction $y \\colon I\\subset \\mathbb{R} \\to \\mathbb{R}$ définie sur un intervalle $I=[t_0,T]$ telle que\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", "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", "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* obtenue **en construisant une suite récurrente**.\n", "\n", "
\n", "\n", "Les schémas que l'on va construire permettent de calculer (explicitement ou implicitement) $u_{n+1}$ à partir de $u_n, u_{n-1}, ..., u_{n-k}$ et il est donc possible de calculer successivement $u_1$, $u_2$,..., en partant de $u_0$ par une formule de récurrence de la forme\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "\\vdots\\\\\n", "u_\\kappa\\approx y_\\kappa,\\\\\n", "u_{n+1}=\\Phi(u_{n+1},u_n, u_{n-1}, \\dots, u_{n-k}),&\\forall n=\\kappa,\\kappa+1,\\dots,N-1.\n", "\\end{cases}$$\n", " \n", "Plus précisement, **nous considérerons des schémas à $k=p+1$ étapes (=pas) linéaires** qui s'écrivent sous la forme générale\n", "$$\n", "u_{n+1} = \\sum_{j=0}^p a_ju_{n-j} + h\\sum_{j=0}^p b_j\\varphi(t_{n-j},u_{n-j}) + hb_{-1}\\varphi(t_{n+1},u_{n+1}), \\qquad\n", "n=p,p+1,\\dots,N-1\n", "$$\n", "où les $\\{a_k\\}$ et $\\{b_k\\}$ sont des coefficients donnés et $p\\ge0$ un entier.\n", "\n", "
\n", "\n", "\n", "**Méthodes explicites et méthodes implicites** \n", "Une méthode est dite *explicite* si la valeur $u_{n+1}$ peut être calculée directement à l'aide des valeurs précédentes $u_k$, $k\\le n$ (ou d'une partie d'entre elles). Pour un schéma linéaire cela équivaut à $b_{-1}=0$. \n", "Une méthode est dite *implicite* si $u_{n+1}$ n'est défini que par une relation implicite faisant intervenir la fonction $\\varphi$. Pour un schéma linéaire cela équivaut à $b_{-1}\\neq0$.\n", "\n", "\n", "**Méthodes à un pas et méthodes multi-pas** \n", "Une méthode numérique pour l'approximation du problème de Cauchy est dite *à un pas* si pour tout $n\\in\\mathbb{N}$, $u_{n+1}$ ne dépend que de $u_n$ et éventuellement de lui-même. \n", "Autrement, on dit que le schéma est une méthode *multi-pas* (ou à pas multiples). \n", "\n", "Les schémas d'Euler explicite, d'Euler implicite et de Crank-Nicolson sont des schémas à un pas ($p=0$) linéaires \n", "\n", "$$ u_{n+1} = a_0u_{n}+hb_0\\varphi(t_{n},u_{n})+hb_{-1}\\varphi(t_{n+1},u_{n+1}),$$\n", "\n", "avec\n", "\n", "| \t| \t| Type \t| $a_0$ \t| $b_0$ \t| $b_{-1}$ \t|\n", "|---\t|---\t|---\t|---\t|---\t|---\t|\n", "| EE \t| $u_{n+1}=u_n+h\\varphi(t_n,u_n)$ \t| Explicite \t| 1 \t| 1 \t| 0 \t|\n", "| EI \t| $u_{n+1}=u_n+h\\varphi(t_{n+1},u_{n+1})$ \t| Implicite \t| 1 \t| 0 \t| 1 \t|\n", "| CN \t| $u_{n+1}=u_n+\\frac{h}{2}\\left(\\varphi(t_n,u_n)+\\varphi(t_{n+1},u_{n+1})\\right)$ \t| Implicite \t| 1 \t| $$\\tfrac{1}{2}$$ \t| $$\\tfrac{1}{2}$$ \t|" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
Remarques pour l'implémentation\n", " \n", "1. `range(5)` produit 0,1,2,3,4 \n", " `range(1,5)` produit 1,2,3,4 \n", " \n", "2. $\\{t_0, t_1,\\dots , t_{N} \\}$ $N+1$ points de *discrétisation* \n", " $\\{u_0, u_1,\\dots , u_{N} \\}$ $N+1$ valeurs à calculer.\n", " \n", "3. `len(tt)` $=N+1$\n", "\n", "Pour un schéma à un pas, $u_{n+1}=F(u_n)$ donc on initialise $u_0$ et on calcule $u_{n+1}$ pour $n$ de $0$ jusqu'à $N-1$, autrement dit `n in range(N)` soit encore `n in range(len(tt)-1)`\n", "\n", "Pour un schéma à deux pas, $u_{n+1}=F(u_n,u_{n-1})$ donc on initialise $u_0$ et $u_1$ et on calcule $u_{n+1}$ pour $n$ de $1$ jusqu'à $N-1$, autrement dit `n in range(1,N)` soit encore `n in range(1,len(tt)-1)`\n", "\n", "Pour un schéma à trois pas, $u_{n+1}=F(u_n,u_{n-1},u_{n-2})$ donc on initialise $u_0$, $u_1$ et $u_2$ et on calcule $u_{n+1}$ pour $n$ de $2$ jusqu'à $N-1$, autrement dit `n in range(2,N)` soit encore `n in range(2,len(tt)-1)`\n", "\n", "...\n", "\n", "\n", "Pour un schéma à $q$ pas, $u_{n+1}=F(u_n,u_{n-1},u_{n-2},\\dots,u_{n-q+1})$ donc on initialise $u_0$, $u_1$ et $u_{q-1}$ et on calcule $u_{n+1}$ pour $n$ de $q-1$ jusqu'à $N-1$, autrement dit `n in range(q-1,N)` soit encore `n in range(q-1,len(tt)-1)`\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dans la suite nous allons écrire explicitement ces schémas.\n", "Pour vérifier nos calculs d'interpolation puis intégration, nous allons utiliser le package de calcul formel `SymPy`." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import sympy as symb\n", "symb.init_printing()\n", "\n", "symb.var('h,t,t_n')\n", "t_np1 = t_n + h\n", "t_nm1 = t_n - h\n", "t_nm2 = t_n - 2 * h\n", "t_nm3 = t_n - 3 * h\n", "t_nm4 = t_n - 4 * h\n", "t_nm5 = t_n - 5 * h\n", "#\n", "y_np1=symb.Symbol('y_{n+1}')\n", "y_n =symb.Symbol('y_{n}')\n", "y_nm1=symb.Symbol('y_{n-1}')\n", "y_nm2=symb.Symbol('y_{n-2}')\n", "y_nm3=symb.Symbol('y_{n-3}')\n", "y_nm4=symb.Symbol('y_{n-4}')\n", "y_nm5=symb.Symbol('y_{n-5}')\n", "y_np1=symb.Symbol('y_{n+1}')\n", "# Pour Adam et N-MS\n", "phi_np1=symb.Symbol('\\phi_{n+1}')\n", "phi_n =symb.Symbol('\\phi_{n}')\n", "phi_nm1=symb.Symbol('\\phi_{n-1}')\n", "phi_nm2=symb.Symbol('\\phi_{n-2}')\n", "phi_nm3=symb.Symbol('\\phi_{n-3}')\n", "phi_nm4=symb.Symbol('\\phi_{n-4}')\n", "phi_nm5=symb.Symbol('\\phi_{n-5}')\n", "# Pour BDF\n", "p = symb.Function('p')\n", "u_np1=symb.Symbol('u_{n+1}')\n", "u_n =symb.Symbol('u_{n}')\n", "u_nm1=symb.Symbol('u_{n-1}')\n", "u_nm2=symb.Symbol('u_{n-2}')\n", "u_nm3=symb.Symbol('u_{n-3}')\n", "u_nm4=symb.Symbol('u_{n-4}')\n", "u_nm5=symb.Symbol('u_{n-5}')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Schémas d'Adam\n", "\n", "Si nous intégrons l'EDO $y'(t)=\\varphi(t,y(t))$ entre $t_n$ et $t_{n+1}$ nous obtenons\n", "$$\n", "y_{n+1}-y_n=\\int_{t_n}^{t_{n+1}} \\varphi(t,y(t))dt.\n", "$$\n", "On peut construire différents schémas selon la formule de quadrature utilisée pour approcher le membre de droite. \n", "Cette solution approchée sera obtenue en construisant une suite récurrente comme suit:\n", "
\n", "\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1}=u_n+\\displaystyle\\int_{t_n}^{t_{n+1}} \\tilde f(t) \\mathrm{d}t \n", "\\quad \\text{où $\\tilde f(t)$ est un polynôme interpolant }\\varphi(t,y(t)) \n", "\\end{cases}\n", "$$\n", "\n", "
\n", "\n", "Les **schémas d'Adam** approchent l'intégrale $\\int_{t_n}^{t_{n+1}} \\varphi(t,y(t))\\mathrm{d}t$ par l'intégrale d'un polynôme $p$ **interpolant $\\varphi$ en des points donnés qui peuvent être à l'extérieur de l'intervalle d'intégration** $[t_{n};t_{n+1}]$. \n", "On peut construire différents schémas selon les points d'interpolation choisis. \n", "\n", "Toutes ces méthodes peuvent s'écrire sous la forme\n", "
\n", "\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_1\\approx y_1,\\\\\n", "\\vdots\\\\\n", "u_{q-1}\\approx y_{q-1},\\\\\n", "\\displaystyle u_{n+1}=u_n+h\\sum_{j=n-q+1}^{n}b_j\\varphi(t_j,u_j)+hb_{-1}\\varphi(t_{n+1},u_{n+1})\n", "\\qquad n=q-1,q,\\dots,N-1\n", "\\end{cases}\n", "$$\n", "\n", "
\n", "\n", "Ils se divisent en deux familles:\n", "+ les schémas d'**Adam-Bashforth** à $q$ pas (AB-$q$) sont \n", " - explicites, \n", " - d'ordre $q$, \n", " - approchent l'intégrale $\\int_{t_n}^{t_{n+1}} \\varphi(t,y(t))\\mathrm{d}t$ par l'intégrale $\\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t$ où $p$ est le polynôme interpolant $\\varphi$ en \n", "$$\n", "\\{t_n,t_{n-1},\\dots,t_{n-q+1}\\} \\text{ avec } q\\ge1;\n", "$$\n", "+ les schémas d'**Adam-Moulton** à $q$ pas (AM-$q$) sont \n", " - implicites, \n", " - d'ordre $q+1$,\n", " - approchent l'intégrale $\\int_{t_n}^{t_{n+1}} \\varphi(t,y(t))\\mathrm{d}t$ par l'intégrale $\\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t$ où $p$ est le polynôme interpolant $\\varphi$ en \n", "$$\n", "\\{t_{n+1},t_n,t_{n-1},\\dots,t_{n-q+1}\\} \\text{ avec } q\\ge0.\n", "$$\n", "\n", "**Remarques** \n", "1. La notation AM-$k$ n'est pas toujours la même selon les auteurs. Par exemple, dans les livres de *Quarteroni et al.*, le $k$ indique l'ordre de la méthode et non le nombre de pas. Les deux notations coincident pour les methodes AB mais pas pour les méthodes AM.\n", "2. Notons que, pour calculer successivement $u_{q}$, $u_{q+1}$, ..., il faut d'abord initialiser $u_0,u_1,\\dots,u_{q-1}$. \n", " Cette initialisation doit être faite par des approximations adéquates qui ne dégradent pas l'ordre de la méthode. \n", "3. On verra plus tard que \n", " - la condition de zéro-stabilité est toujours satisfaite, \n", " - la condition de A-stabilité est de plus en plus contraignante quand l’ordre augmente. Plus particulièrement, une méthode explicite A-stable a au plus ordre 2 (seconde barrière de Dahlquist)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AB-1\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_n,y(t_n))$\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =h\\varphi(t_n,y(t_n))$\n", "et on obtient le schéma\n", "
\n", "\n", "$$\n", "\\text{(AB$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_{n}+h \\varphi(t_{n},u_{n})& n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "\n", "
\n", "La méthode AB$_1$ coïncide avec la méthode d'Euler progressive. " ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\phi_{n} h + u_{n}$" ], "text/plain": [ "u_{n+1} = \\phi_{n}⋅h + u_{n}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n)], t)\n", "#display(pol)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AB-2\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n));(t_{n-1},\\varphi(t_{n-1},y_{n-1}))\\}$\n", "+ Polynôme: \n", "$p(t)=\\varphi(t_n,y_n)\\frac{t-t_{n-1}}{t_n-t_{n-1}}+\\varphi(t_{n-1},y(t_{n-1}))\\frac{t-t_n}{t_{n-1}-t_n}\n", " =\\varphi(t_n,y_n)\\frac{t-t_{n-1}}{h}+\\varphi(t_{n-1},y(t_{n-1}))\\frac{t-t_n}{-h}$\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{2}\\left(3\\varphi(t_n,y(t_n))-\\varphi(t_{n-1},y(t_{n-1}))\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\\text{(AB$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n}+ \\dfrac{h}{2} \\big(3\\varphi(t_{n},u_{n})-\\varphi(t_{n-1},u_{n-1})\\big)& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AB$_1$. \n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- \\phi_{n-1} + 3 \\phi_{n}\\right)}{2} + u_{n}$" ], "text/plain": [ " h⋅(-\\phi_{n-1} + 3⋅\\phi_{n}) \n", "u_{n+1} = ──────────────────────────── + u_{n}\n", " 2 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n), (t_nm1, phi_nm1)], t)\n", "# display(pol)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AB-3\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n));(t_{n-1},\\varphi(t_{n-1},y_{n-1}));(t_{n-2},\\varphi(t_{n-2},y_{n-2}))\\}$\n", "+ Polynôme: \n", "\\begin{align}\n", "p(t)&=\\varphi(t_n,y_n)\\frac{(t-t_{n-1})(t-t_{n-2})}{(t_n-t_{n-1})(t_n-t_{n-2})}\n", " +\\varphi(t_{n-1},y(t_{n-1}))\\frac{(t-t_{n})(t-t_{n-2})}{(t_{n-1}-t_{n})(t_{n-1}n-t_{n-2})}\n", " +\\varphi(t_{n-2},y(t_{n-2}))\\frac{(t-t_{n})(t-t_{n-1})}{(t_{n-2}-t_{n})(t_{n-2}n-t_{n-1})}\n", " \\\\\n", " &=\\frac{\\varphi(t_{n-2},y_{n-2})}{2h^2}(t-t_{n-1})(t-t_{n})\n", " -\\frac{\\varphi(t_{n-1},y_{n-1})}{h^2}(t-t_{n-2})(t-t_{n})\n", " +\\frac{\\varphi(t_{n},y_{n})}{2h^2}(t-t_{n-2})(t-t_{n-1})\n", " \\end{align}\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{12}\\left(23\\varphi(t_n,y(t_n))-16\\varphi(t_{n-1},y(t_{n-1}))+5\\varphi(t_{n-2},y(t_{n-2}))\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(AB$_3$)}\\quad\n", "\\begin{cases}\n", "u_0 = y(t_0) = y_0,\\\\\n", "u_1 = u_0 + h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_2 = u_1 + \\dfrac{h}{2} \\big(3\\varphi(t_1,u_1)-\\varphi(t_0,u_0)\\big)\\approx y(t_2)\\\\\n", "u_{n+1} = u_{n} + \\dfrac{h}{12}\\big(23\\varphi(t_n,u_n)-16\\varphi(t_{n-1},u_{n-1})+5\\varphi(t_{n-2},u_{n-2})\\big) & n=2,3,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AB$_1$ et $u_{2}$ est une approximation de $y(t_2)$ obtenue en utilisant la méthode AB$_2$.\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 16 \\phi_{n-1} + 5 \\phi_{n-2} + 23 \\phi_{n}\\right)}{12} + u_{n}$" ], "text/plain": [ " h⋅(-16⋅\\phi_{n-1} + 5⋅\\phi_{n-2} + 23⋅\\phi_{n}) \n", "u_{n+1} = ─────────────────────────────────────────────── + u_{n}\n", " 12 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2)], t)\n", "# display(pol)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AB-4\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n));(t_{n-1},\\varphi(t_{n-1},y_{n-1}));(t_{n-2},\\varphi(t_{n-2},y_{n-2}));(t_{n-3},\\varphi(t_{n-3},y_{n-3}))\\}$\n", "+ Polynôme: $p(t)=\\displaystyle\\sum_{i=n-3}^{n}\\left(\\varphi(t_i,y_i)\\prod_{\\substack{j=n-2,...,n\\\\j\\neq i}} \\frac{t-t_j}{t_i-t_j}\\right)$\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{24}\\left(55\\varphi(t_n,y(t_n))-59\\varphi(t_{n-1},y(t_{n-1}))+37\\varphi(t_{n-2},y(t_{n-2})-9\\varphi(t_{n-3},y(t_{n-3}))\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(AB$_4$)}\\quad\n", "\\begin{cases}\n", "u_0 = y(t_0) = y_0,\\\\\n", "u_1 = u_0 + h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_2 = u_1 + \\dfrac{h}{2} \\big(3\\varphi(t_1,u_1)-\\varphi(t_0,u_0)\\big)\\approx y(t_2)\\\\\n", "u_{3} = u_{2} + \\dfrac{h}{12}\\big(23\\varphi(t_2,u_2)-16\\varphi(t_{1},u_{1})+5\\varphi(t_{0},u_{0})\\big)\\approx y(t_3) \\\\\n", "u_{n+1} = u_{n} + \\dfrac{h}{24}\\big(55\\varphi(t_n,u_n)-59\\varphi(t_{n-1},u_{n-1})+37\\varphi(t_{n-2},u_{n-2})-9\\varphi(t_{n-3},u_{n-3})\\big) & n=3,4,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AB$_1$, $u_{2}$ est une approximation de $y(t_2)$ obtenue en utilisant la méthode AB$_2$ etc.\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 59 \\phi_{n-1} + 37 \\phi_{n-2} - 9 \\phi_{n-3} + 55 \\phi_{n}\\right)}{24} + u_{n}$" ], "text/plain": [ " h⋅(-59⋅\\phi_{n-1} + 37⋅\\phi_{n-2} - 9⋅\\phi_{n-3} + 55⋅\\phi_{n}) \n", "u_{n+1} = ─────────────────────────────────────────────────────────────── + u_\n", " 24 \n", "\n", " \n", "{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2),\n", " (t_nm3, phi_nm3)], t)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour la construction des autres schémas AB$_5$ et AB$_6$ on verra plus bas dans le récapitulatif." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AM-0\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_{n+1},\\varphi(t_{n+1},y_{n+1}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_{n+1},y(t_{n+1}))$\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =h\\varphi(t_{n+1},y(t_{n+1}))$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(AM$_0$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_{n}+h \\varphi(t_{n+1},u_{n+1})& n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "\n", "
\n", "La méthode AM$_0$ coïncide avec la méthode d'Euler regressive. " ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\phi_{n+1} h + u_{n}$" ], "text/plain": [ "u_{n+1} = \\phi_{n+1}⋅h + u_{n}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, phi_np1)], t)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AM-1\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_{n+1},\\varphi(t_{n+1},y_{n+1})),(t_{n},\\varphi(t_{n},y_{n}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{t-t_n}{t_{n+1}-t_n}+\\varphi(t_{n},y_{n})\\frac{t-t_{n+1}}{t_n-t_{n+1}}$\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{2}\\left(\\varphi(t_{n+1},y_{n+1})+\\varphi(t_n,y_n)\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(AM$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_{n}+ \\dfrac{h}{2} \\big(\\varphi(t_{n},u_{n})+\\varphi(t_{n+1},u_{n+1})\\big)& n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "\n", "
\n", "La méthode AM$_1$ coïncide avec la méthode de Crank-Nicolson. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(\\phi_{n+1} + \\phi_{n}\\right)}{2} + u_{n}$" ], "text/plain": [ " h⋅(\\phi_{n+1} + \\phi_{n}) \n", "u_{n+1} = ───────────────────────── + u_{n}\n", " 2 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, phi_np1), (t_n, phi_n)], t)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### AM-2\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_{n+1},\\varphi(t_{n+1},y_{n+1})),(t_{n},\\varphi(t_{n},y_{n})),(t_{n-1},\\varphi(t_{n-1},y_{n-1}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{(t-t_n)(t-t_{n-1})}{(t_{n+1}-t_n)(t_{n+1}-t_{n-1})} +\\varphi(t_{n},y_{n})\\frac{(t-t_{n+1})(t-t_{n-1})}{(t_n-t_{n+1})(t_n-t_{n-1})}+\\varphi(t_{n-1},y_{n-1})\\frac{(t-t_{n+1})(t-t_{n})}{(t_{n-1}-t_{n+1})(t_{n+1}-t_{n})}$\n", "+ Approximation: $y_{n+1}-y_n \\approx \\int_{t_n}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{12}\\left(5\\varphi(t_{n+1},y_{n+1})+8\\varphi(t_n,y_n)-\\varphi(t_{n-1},y_{n-1}))\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(AM$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_{0}+ \\dfrac{h}{2} \\big(\\varphi(t_{1},u_{1})+\\varphi(t_{0},u_{0})\\big)\\\\\n", "u_{n+1}=u_{n}+\\dfrac{h}{12}\\big(5\\varphi(t_{n+1},u_{n+1})+8\\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1})\\big) & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AM$_1$. \n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(5 \\phi_{n+1} - \\phi_{n-1} + 8 \\phi_{n}\\right)}{12} + u_{n}$" ], "text/plain": [ " h⋅(5⋅\\phi_{n+1} - \\phi_{n-1} + 8⋅\\phi_{n}) \n", "u_{n+1} = ────────────────────────────────────────── + u_{n}\n", " 12 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, phi_np1), (t_n, phi_n), (t_nm1, phi_nm1)], t)\n", "integrale = symb.integrate(pol, (t, t_n, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_n + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour la construction des autres schémas AM$_3$, AM$_4$, AM$_5$ et AM$_6$ on verra plus bas dans le récapitulatif." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calcul systématique des coefficients des méthodes AB et AM" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "AB-1: explicite, à 1 pas, d'ordre 1\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\phi_{n} h + u_{n}$" ], "text/plain": [ "u_{n+1} = \\phi_{n}⋅h + u_{n}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AB-2: explicite, à 2 pas, d'ordre 2\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- \\phi_{n-1} + 3 \\phi_{n}\\right)}{2} + u_{n}$" ], "text/plain": [ " h⋅(-\\phi_{n-1} + 3⋅\\phi_{n}) \n", "u_{n+1} = ──────────────────────────── + u_{n}\n", " 2 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AB-3: explicite, à 3 pas, d'ordre 3\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 16 \\phi_{n-1} + 5 \\phi_{n-2} + 23 \\phi_{n}\\right)}{12} + u_{n}$" ], "text/plain": [ " h⋅(-16⋅\\phi_{n-1} + 5⋅\\phi_{n-2} + 23⋅\\phi_{n}) \n", "u_{n+1} = ─────────────────────────────────────────────── + u_{n}\n", " 12 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AB-4: explicite, à 4 pas, d'ordre 4\n" ] }, { "data": { "image/png": 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c7mgIrIUA+738Ibn7Al10U05Dm7k5AsSihsDaCJjyXRtxu58hsAICUq6YlFuHqJTO6WczN6/wDOwWhkAXAqZ8u9CxPENghwh4BXupsPYxG11jbubdQzM37/C5GsmnhcDqrxqdFnzWG0NgWwhIsfJa0SOF1T6vV7oQivJNHXikDq8nhRPTVV2lmTMEDIEFEFj9tPMCfbAmDQFDQAhIyXLAqqZ4PTAo15fKZ8Xb+tCA0nlH+K1Ce8/XA2aBIbA0AqZ8l0bY2jcEVkBAipNVLQesUKLNPd3HSmseuoqp4h3g1HvAcRmLGwKGwIwImPKdEUxryhA4IgL8UxgKmANVTVd7xzdkeiVNHRwKGuX9rkdRu8L2YwgYAtMQ+D+GC503oKcb3wAAAABJRU5ErkJggg==\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 59 \\phi_{n-1} + 37 \\phi_{n-2} - 9 \\phi_{n-3} + 55 \\phi_{n}\\right)}{24} + u_{n}$" ], "text/plain": [ " h⋅(-59⋅\\phi_{n-1} + 37⋅\\phi_{n-2} - 9⋅\\phi_{n-3} + 55⋅\\phi_{n}) \n", "u_{n+1} = ─────────────────────────────────────────────────────────────── + u_\n", " 24 \n", "\n", " \n", "{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AB-5: explicite, à 5 pas, d'ordre 5\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 2774 \\phi_{n-1} + 2616 \\phi_{n-2} - 1274 \\phi_{n-3} + 251 \\phi_{n-4} + 1901 \\phi_{n}\\right)}{720} + u_{n}$" ], "text/plain": [ " h⋅(-2774⋅\\phi_{n-1} + 2616⋅\\phi_{n-2} - 1274⋅\\phi_{n-3} + 251⋅\\phi_{\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 720 \n", "\n", "n-4} + 1901⋅\\phi_{n}) \n", "───────────────────── + u_{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AB-6: explicite, à 6 pas, d'ordre 6\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 7923 \\phi_{n-1} + 9982 \\phi_{n-2} - 7298 \\phi_{n-3} + 2877 \\phi_{n-4} - 475 \\phi_{n-5} + 4277 \\phi_{n}\\right)}{1440} + u_{n}$" ], "text/plain": [ " h⋅(-7923⋅\\phi_{n-1} + 9982⋅\\phi_{n-2} - 7298⋅\\phi_{n-3} + 2877⋅\\phi_\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 1440 \n", "\n", "{n-4} - 475⋅\\phi_{n-5} + 4277⋅\\phi_{n}) \n", "─────────────────────────────────────── + u_{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "\n", "\n", "AM-0: implicite, à 0 pas, d'ordre 1\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\phi_{n+1} h + u_{n}$" ], "text/plain": [ "u_{n+1} = \\phi_{n+1}⋅h + u_{n}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AM-1: implicite, à 1 pas, d'ordre 2\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(\\phi_{n+1} + \\phi_{n}\\right)}{2} + u_{n}$" ], "text/plain": [ " h⋅(\\phi_{n+1} + \\phi_{n}) \n", "u_{n+1} = ───────────────────────── + u_{n}\n", " 2 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AM-2: implicite, à 2 pas, d'ordre 3\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(5 \\phi_{n+1} - \\phi_{n-1} + 8 \\phi_{n}\\right)}{12} + u_{n}$" ], "text/plain": [ " h⋅(5⋅\\phi_{n+1} - \\phi_{n-1} + 8⋅\\phi_{n}) \n", "u_{n+1} = ────────────────────────────────────────── + u_{n}\n", " 12 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AM-3: implicite, à 3 pas, d'ordre 4\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(9 \\phi_{n+1} - 5 \\phi_{n-1} + \\phi_{n-2} + 19 \\phi_{n}\\right)}{24} + u_{n}$" ], "text/plain": [ " h⋅(9⋅\\phi_{n+1} - 5⋅\\phi_{n-1} + \\phi_{n-2} + 19⋅\\phi_{n}) \n", "u_{n+1} = ────────────────────────────────────────────────────────── + u_{n}\n", " 24 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AM-4: implicite, à 4 pas, d'ordre 5\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(251 \\phi_{n+1} - 264 \\phi_{n-1} + 106 \\phi_{n-2} - 19 \\phi_{n-3} + 646 \\phi_{n}\\right)}{720} + u_{n}$" ], "text/plain": [ " h⋅(251⋅\\phi_{n+1} - 264⋅\\phi_{n-1} + 106⋅\\phi_{n-2} - 19⋅\\phi_{n-3} \n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 720 \n", "\n", "+ 646⋅\\phi_{n}) \n", "─────────────── + u_{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AM-5: implicite, à 5 pas, d'ordre 6\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(475 \\phi_{n+1} - 798 \\phi_{n-1} + 482 \\phi_{n-2} - 173 \\phi_{n-3} + 27 \\phi_{n-4} + 1427 \\phi_{n}\\right)}{1440} + u_{n}$" ], "text/plain": [ " h⋅(475⋅\\phi_{n+1} - 798⋅\\phi_{n-1} + 482⋅\\phi_{n-2} - 173⋅\\phi_{n-3}\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 1440 \n", "\n", " + 27⋅\\phi_{n-4} + 1427⋅\\phi_{n}) \n", "───────────────────────────────── + u_{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "AM-6: implicite, à 6 pas, d'ordre 7\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(19087 \\phi_{n+1} - 46461 \\phi_{n-1} + 37504 \\phi_{n-2} - 20211 \\phi_{n-3} + 6312 \\phi_{n-4} - 863 \\phi_{n-5} + 65112 \\phi_{n}\\right)}{60480} + u_{n}$" ], "text/plain": [ " h⋅(19087⋅\\phi_{n+1} - 46461⋅\\phi_{n-1} + 37504⋅\\phi_{n-2} - 20211⋅\\p\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 60480 \n", "\n", "hi_{n-3} + 6312⋅\\phi_{n-4} - 863⋅\\phi_{n-5} + 65112⋅\\phi_{n}) \n", "───────────────────────────────────────────────────────────── + u_{n}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n" ] } ], "source": [ "Points = [(t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2), (t_nm3, phi_nm3),\n", " (t_nm4, phi_nm4), (t_nm5, phi_nm5)]\n", "for q in range(1, len(Points) + 1):\n", " pol = symb.interpolate(Points[:q], t)\n", " AB = (symb.integrate(pol, (t, t_n, t_np1)).simplify())\n", " print(f\"AB-{q}: explicite, à {q} pas, d'ordre {q}\")\n", " display(symb.Eq(u_np1, u_n + AB))\n", " print(\"\\n\")\n", "\n", "print(\"\\n\")\n", "\n", "Points = [(t_np1, phi_np1), (t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2),\n", " (t_nm3, phi_nm3), (t_nm4, phi_nm4), (t_nm5, phi_nm5)]\n", "for q in range(len(Points)):\n", " pol = symb.interpolate(Points[0:q + 1], t)\n", " AM = (symb.integrate(pol, (t, t_n, t_np1)).simplify())\n", " print(f\"AM-{q}: implicite, à {q} pas, d'ordre {q+1}\")\n", " display(symb.Eq(u_np1, u_n + AM))\n", " print(\"\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Schémas de Nyström et de Milne-Simpson\n", "\n", "Les méthodes d'Adam peuvent être facilement généralisées en intégrant l'EDO $y'(t)=\\varphi(t,y(t))$ entre $t_{n-r}$ et $t_{n+1}$ avec $r\\ge1$. \n", "\n", "Avec $r=1$, si nous intégrons l'EDO $y'(t)=\\varphi(t,y(t))$ entre $t_{n-1}$ et $t_{n+1}$ nous obtenons\n", "$$\n", "y_{n+1}-y_{n-1}=\\int_{t_{n-1}}^{t_{n+1}} \\varphi(t,y(t))dt.\n", "$$\n", "On peut construire différents schémas selon la formule de quadrature utilisée pour approcher le membre de droite. \n", " \n", "Cette solution approchée sera obtenue en construisant une suite récurrente comme suit:\n", "
\n", "\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_1=?,\\\\\n", "u_{n+1}=u_{n-1}+\\displaystyle\\int_{t_{n-1}}^{t_{n+1}} \\tilde f(t) \\mathrm{d}t \n", "\\quad \\text{où $\\tilde f(t)$ est un polynôme interpolant }\\varphi(t,y(t)) \n", "\\end{cases}\n", "$$\n", "\n", "
\n", "\n", "Ces schémas approchent l'intégrale $\\int_{t_{n-1}}^{t_{n+1}} \\varphi(t,y(t))\\mathrm{d}t$ par l'intégrale d'un polynôme $p$ **interpolant $\\varphi$ en des points donnés qui peuvent être à l'extérieur de l'intervalle d'intégration** $[t_{n-1};t_{n+1}]$. \n", "On peut construire différents schémas selon les points d'interpolation choisis. \n", "Toutes ces méthodes peuvent s'écrire à nouveau sous la forme\n", "
\n", "\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_1\\approx y_1,\\\\\n", "\\vdots\\\\\n", "u_{q-1}\\approx y_{q-1},\\\\\n", "\\displaystyle u_{n+1}=u_{n-1}+h\\sum_{j=n-q+1}^{n}b_j\\varphi(t_j,u_j)+hb_{-1}\\varphi(t_{n+1},u_{n+1})\n", "\\qquad n=q-1,q,\\dots,N-1.\n", "\\end{cases}\n", "$$\n", "\n", "
\n", "\n", "Ils se divisent en deux familles:\n", "+ les schémas de **Nyström** à $q$ pas (N-$q$) sont \n", " - explicites, \n", " - approchent l'intégrale $\\int_{t_{n-1}}^{t_{n+1}} \\varphi(t,y(t))\\mathrm{d}t$ par l'intégrale $\\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t$ où $p$ est le polynôme interpolant $\\varphi$ en \n", "$$\n", "\\{t_n,t_{n-1},\\dots,t_{n-q+1}\\} \\text{ avec } q\\ge1.\n", "$$\n", "+ les schémas de **Milne-Simpson** à $q$ pas (MS-$q$) sont \n", " - implicites, \n", " - approchent l'intégrale $\\int_{t_{n-1}}^{t_{n+1}} \\varphi(t,y(t))\\mathrm{d}t$ par l'intégrale $\\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t$ où $p$ est le polynôme interpolant $\\varphi$ en \n", "$$\n", "\\{t_{n+1},t_n,t_{n-1},\\dots,t_{n-q+1}\\} \\text{ avec } q\\ge0.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### N-1\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_n,y(t_n))$\n", "+ Approximation: $y_{n+1}-y_{n-1} \\approx \\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t =2h\\varphi(t_n,y(t_n))$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(N$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n},u_{n})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler explicite. \n", "\n", "
\n", "\n", "La méthode N$_1$ coïncide avec la **méthode du point milieu** (appelée aussi **Saute-mouton** ou **Leapfrog**). \n", "\n", "Remarque: on l'appelle N$_1$ mais il est à 2 pas car il dépend de $u_{n-1}$." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n)], t).simplify()\n", "integrale = symb.integrate(pol, (t, t_nm1, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_nm1 + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### N-2\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n)),(t_{n-1},\\varphi(t_{n-1},y_{n-1}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_n,y(t_n))\\frac{t-t_{n-1}}{t_n-t_{n-1}}+\\varphi(t_{n-1},y_{n-1})\\frac{t-t_n}{t_{n-1}-t_n}$\n", "+ Approximation: $y_{n+1}-y_{n-1} \\approx \\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t =2h\\varphi(t_n,y(t_n))$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(N$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n},u_{n})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler explicite. \n", "\n", "
\n", "On retrouve la méthode N$_1$." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n), (t_nm1, phi_nm1)], t).simplify()\n", "integrale = symb.integrate(pol, (t, t_nm1, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_nm1 + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### N-3\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_n,\\varphi(t_n,y_n));(t_{n-1},\\varphi(t_{n-1},y_{n-1}));(t_{n-2},\\varphi(t_{n-2},y_{n-2}))\\}$\n", "+ Polynôme: \n", "$\\begin{aligned}[t]\n", " p(t)&=\\varphi(t_n,y_n)\\frac{(t-t_{n-1})(t-t_{n-2})}{(t_n-t_{n-1})(t_n-t_{n-2})}\n", " +\\varphi(t_{n-1},y_{n-1})\\frac{(t-t_{n})(t-t_{n-2})}{(t_{n-1}-t_{n})(t_{n-1}n-t_{n-2})}\n", " +\\varphi(t_{n-2},y_{n-2})\\frac{(t-t_{n})(t-t_{n-1})}{(t_{n-2}-t_{n})(t_{n-2}n-t_{n-1})}\n", " \\\\\n", " &=\\frac{\\varphi(t_{n-2},y_{n-2})}{2h^2}(t-t_{n-1})(t-t_{n})\n", " -\\frac{\\varphi(t_{n-1},y_{n-1})}{h^2}(t-t_{n-2})(t-t_{n})\n", " +\\frac{\\varphi(t_{n},y_{n})}{2h^2}(t-t_{n-2})(t-t_{n-1})\n", " \\end{aligned}$\n", "+ Approximation: $y_{n+1}-y_{n-1} \\approx \\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{3}\\left(7\\varphi(t_n,y_n)-2\\varphi(t_{n-1},y_{n-1})+\\varphi(t_{n-2},y_{n-2})\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(N$_3$)}\\quad\n", "\\begin{cases}\n", "u_0 = y(t_0) = y_0,\\\\\n", "u_1 = u_0 + h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_2 = u_{0} + 2h\\varphi(t_{1},u_{1})\\approx y(t_2)\\\\\n", "u_{n+1} = u_{n-1} + \\dfrac{h}{3}\\big(7\\varphi(t_n,u_n)-2\\varphi(t_{n-1},u_{n-1})+\\varphi(t_{n-2},u_{n-2})\\big) & n=2,3,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler explicite et $u_{2}$ est une approximation de $y(t_2)$ obtenue en utilisant la méthode N$_2$. \n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 2 \\phi_{n-1} + \\phi_{n-2} + 7 \\phi_{n}\\right)}{3} + u_{n-1}$" ], "text/plain": [ " h⋅(-2⋅\\phi_{n-1} + \\phi_{n-2} + 7⋅\\phi_{n}) \n", "u_{n+1} = ─────────────────────────────────────────── + u_{n-1}\n", " 3 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2)],\n", " t).simplify()\n", "integrale = symb.integrate(pol, (t, t_nm1, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_nm1 + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour la construction des autres schémas N$_4$, N$_5$ et N$_6$ on verra plus bas dans le récapitulatif." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### MS-0\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_{n+1},\\varphi(t_{n+1},y_{n+1}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_{n+1},y(t_{n+1}))$\n", "+ Approximation: $y_{n+1}-y_{n-1} \\approx \\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t =2h\\varphi(t_{n+1},y(t_{n+1}))$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(MS$_0$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{1},u_{1})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n+1},u_{n+1})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler implicite.\n", "\n", "
\n", "\n", "Remarque: on l'appelle MS$_0$ mais il est à 2 pas car il dépend de $u_{n-1}$." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n+1} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n+1}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, phi_np1)], t).simplify()\n", "integrale = symb.integrate(pol, (t, t_nm1, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_nm1 + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### MS-1\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_{n+1},\\varphi(t_{n+1},y_{n+1})),(t_{n},\\varphi(t_{n},y_{n}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{t-t_n}{t_{n+1}-t_n} +\\varphi(t_{n},y_{n})\\frac{t-t_{n+1}}{t_n-t_{n+1}}$\n", "+ Approximation: $y_{n+1}-y_{n-1} \\approx \\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t =2h\\varphi(t_n,y_n)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(MS$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{1},u_{1})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n},u_{n})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler implicite. \n", "\n", "
\n", "On retrouve la méthode N$_1$: le schéma est devenu explicite.\n", "\n", "Remarque: on l'appelle MS$_1$ mais il est à 2 pas car il dépend de $u_{n-1}$." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, phi_np1), (t_n, phi_n)], t).simplify()\n", "integrale = symb.integrate(pol, (t, t_nm1, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_nm1 + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### MS-2\n", "\n", "On a\n", "+ Points d'interpolation: $\\{(t_{n+1},\\varphi(t_{n+1},y_{n+1})),(t_{n},\\varphi(t_{n},y_{n})),(t_{n-1},\\varphi(t_{n-1},y_{n-1}))\\}$\n", "+ Polynôme: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{(t-t_n)(t-t_{n-1})}{(t_{n+1}-t_n)(t_{n+1}-t_{n-1})}+ \\varphi(t_{n},y_{n})\\frac{(t-t_{n+1})(t-t_{n-1})}{(t_n-t_{n+1})(t_n-t_{n-1})}+ \\varphi(t_{n-1},y_{n-1})\\frac{(t-t_{n+1})(t-t_{n})}{(t_{n-1}-t_{n+1})(t_{n+1}-t_{n})}$\n", "+ Approximation: $y_{n+1}-y_{n-1} \\approx \\int_{t_{n-1}}^{t_{n+1}} p(t) \\mathrm{d}t =\\frac{h}{3}\\left(\\varphi(t_{n+1},y_{n+1})+4\\varphi(t_n,y_n)+\\varphi(t_{n-1},y_{n-1}))\\right)$\n", "et on obtient le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(MS$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_{0}+ \\dfrac{h}{2} \\big(\\varphi(t_{1},u_{1})+\\varphi(t_{0},u_{0})\\big)\\\\\n", "u_{n+1}=u_{n-1}+\\dfrac{h}{3}\\big( \\varphi(t_{n+1},u_{n+1})+4\\varphi(t_n,u_n)+\\varphi(t_{n-1},u_{n-1}\\big) & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AM$_1$ (=Crank-Nicolson). \n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(\\phi_{n+1} + \\phi_{n-1} + 4 \\phi_{n}\\right)}{3} + u_{n-1}$" ], "text/plain": [ " h⋅(\\phi_{n+1} + \\phi_{n-1} + 4⋅\\phi_{n}) \n", "u_{n+1} = ──────────────────────────────────────── + u_{n-1}\n", " 3 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, phi_np1), (t_n, phi_n), (t_nm1, phi_nm1)],\n", " t).simplify()\n", "integrale = symb.integrate(pol, (t, t_nm1, t_np1)).simplify()\n", "display(symb.Eq(u_np1, u_nm1 + integrale))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pour la construction des autres schémas MS$_3$, MS$_4$, MS$_5$ et MS$_6$ on verra plus bas dans le récapitulatif." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calcul systématique des coefficients des méthodes N et MS" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "N-1: explicite, à 1 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "N-2: explicite, à 2 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "N-3: explicite, à 3 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 2 \\phi_{n-1} + \\phi_{n-2} + 7 \\phi_{n}\\right)}{3} + u_{n-1}$" ], "text/plain": [ " h⋅(-2⋅\\phi_{n-1} + \\phi_{n-2} + 7⋅\\phi_{n}) \n", "u_{n+1} = ─────────────────────────────────────────── + u_{n-1}\n", " 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "N-4: explicite, à 4 pas\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 5 \\phi_{n-1} + 4 \\phi_{n-2} - \\phi_{n-3} + 8 \\phi_{n}\\right)}{3} + u_{n-1}$" ], "text/plain": [ " h⋅(-5⋅\\phi_{n-1} + 4⋅\\phi_{n-2} - \\phi_{n-3} + 8⋅\\phi_{n}) \n", "u_{n+1} = ────────────────────────────────────────────────────────── + u_{n-1}\n", " 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "N-5: explicite, à 5 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 266 \\phi_{n-1} + 294 \\phi_{n-2} - 146 \\phi_{n-3} + 29 \\phi_{n-4} + 269 \\phi_{n}\\right)}{90} + u_{n-1}$" ], "text/plain": [ " h⋅(-266⋅\\phi_{n-1} + 294⋅\\phi_{n-2} - 146⋅\\phi_{n-3} + 29⋅\\phi_{n-4}\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 90 \n", "\n", " + 269⋅\\phi_{n}) \n", "──────────────── + u_{n-1}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "N-6: explicite, à 6 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(- 406 \\phi_{n-1} + 574 \\phi_{n-2} - 426 \\phi_{n-3} + 169 \\phi_{n-4} - 28 \\phi_{n-5} + 297 \\phi_{n}\\right)}{90} + u_{n-1}$" ], "text/plain": [ " h⋅(-406⋅\\phi_{n-1} + 574⋅\\phi_{n-2} - 426⋅\\phi_{n-3} + 169⋅\\phi_{n-4\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 90 \n", "\n", "} - 28⋅\\phi_{n-5} + 297⋅\\phi_{n}) \n", "───────────────────────────────── + u_{n-1}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "\n", "\n", "MS-0: implicite, à 0 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n+1} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n+1}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS-1: implicite, à 1 pas\n" ] }, { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAALkAAAAUCAYAAADRL5xsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFWElEQVRoBe2b63EUOxBGF5cDoJzBhQzAZOCbAeAIgAyg7j//c0EGXCLgkQFkwCMDyACXMzDnCGlKI2t2tevZ9di7XSVaarUe3fqm1TNr7lxcXMx2tNgDJycn99B6FTUP4We2kf+IsrkMveeU/+cqNXQyx13UPlLCHmjfbxi21Sr7W219o/EASYC/hf+bhlB/Tf27MsqXJJ/DRwEja52zhmt+h3+bs96uK3pgb+eJJg8I6Be5JiAzqgs4o2oL/W5RWkLnAbqfl9AfVMWWI8rzQYUb3rEDedsBHqH2EyCYKuRkBL+L3Eg/SIKIzqa0ZnCSrCPOp6TlBslGDla1q7RtUPmmdexA3nZigukX4DJy12gRQB4wdixAur5p07z91Pa4tbKQk3MAHpLXlXmjeWbvBYm2V/Iz+NAh0317CbufDFhnyjCj/1KURmbfMcU0xRwaFvzry+pV/ejN8IV5HsMP4ryu9wLZL/jGiPUmj50AcjzyH5vV+TrtHaUDOTLB/xg+dNB0T4PY41t2IgCWoR+r2MYYQWWakr64dGvGfczgIY+HW39DUT+9rK4ERuYQVK4tvaYd5oEbiHx32PQ5TR47+zhHh32lSF6DfhrLSdmlSJUrTKWOLb2XwzXvS1B9Ys03+ToRbGdpL/AQddWhboph2nIVMKaH2Ju1fFB8ADZGrH8jsLOPR3R8AvFT2qeFl3RqKStUxm2yHw/LvXj9Phx39qvPxp68MfRbL2rS9tbTX/9kq5iq5NH+POpkKktVQ9BhznRmabCA+5QaNc6YoZvuQH36a0Fi3k13I7BjJNfpGujhCK48VfF6VTbaSxPr+L255kyWCfvwsNyL5NqTIvYukA/ggq0kwfyBvuDTsjO2ta+MwAOqVbG+6YGZ9Twny/vqiChEr+p35Kap9+C9W2neXPahH+yEbwQ7DfupYmc/G2hU8snMD0jjNaaMGtmwpasexiDFtYweYe1BxUoHY4YiVUW7E82LVJ2Slbin+/AuglMP9sAFrnV/pAmEzIf0Z2zOYltAdONTXwuP412j/D4ueM/pH/OcWraUdDaFnbRelUf7L2EnB7nOK51ktApRnAmMYCFKxbppRPrVzYOTBEA1WvztXu+/61ybuY0Sj+B56qFBPozp9tN/AjuRfslvwfBSzxwhEsP16TJ+NIWbMS6fU5HzhD3EOf3ycpXbwjmXoUljZy+zpOcUnOShHlIS8AWw0SIdYnqbP0TmlwOvuqdwDb5VFG3SXn/4Md3qCjLfG9Lt98x2ZrwPRfAr3PFn8NAPX8WPXdDJ1rDqXClV8Zx6Z1norqPZW4/1J4WdPJIbod6xQd/8/bbrxo0yfqZ6CU9OPKD+gWL0ML1JUYxmcPaZlVtGpgc+vNpcUgoCM3wRUh+4aZNpij8C6U9BeGo/PNEqfnQeH5aSPLtj5j+ilDdNqbuO9qSx04Ec55xjfS1X7MnQS5HpEfoJ+DPkXs1GeucJRN3DLiO7kb/MKT8iyx+WOMM0GHtr/uMqdAWy0V279UfVLuTNfkxeYEztZXeGfKkXxjTfWJz114EdbwMDxCLS172bpBzQgbzsaGgL6tNMz4fBCK/TjWBGtfzqDqrIPlOqhxUUbs8/HlKZO9esW+jH2qCRZYK0C04jz12bbqHN4oeBo+BkJZCzAaOU+Wl+/brxJ8h8EZv7ZNG/DWQ+3vvUVxo9FT+yj5aHsdz+Su3rsHklkGOdIC8P0NTkmOJf65V9iNsoOsEbwIfGb7deWb/h13olt+2+p+V7zSJamx8XLXyN/WuzeQg7dzb9P4PYyLakK9eIo93SuQf28saG6rtUZkOO3i3z1wN/AKdXdsMCbO2pAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle u_{n+1} = 2 \\phi_{n} h + u_{n-1}$" ], "text/plain": [ "u_{n+1} = 2⋅\\phi_{n}⋅h + u_{n-1}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS-2: implicite, à 2 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(\\phi_{n+1} + \\phi_{n-1} + 4 \\phi_{n}\\right)}{3} + u_{n-1}$" ], "text/plain": [ " h⋅(\\phi_{n+1} + \\phi_{n-1} + 4⋅\\phi_{n}) \n", "u_{n+1} = ──────────────────────────────────────── + u_{n-1}\n", " 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS-3: implicite, à 3 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(\\phi_{n+1} + \\phi_{n-1} + 4 \\phi_{n}\\right)}{3} + u_{n-1}$" ], "text/plain": [ " h⋅(\\phi_{n+1} + \\phi_{n-1} + 4⋅\\phi_{n}) \n", "u_{n+1} = ──────────────────────────────────────── + u_{n-1}\n", " 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS-4: implicite, à 4 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(29 \\phi_{n+1} + 24 \\phi_{n-1} + 4 \\phi_{n-2} - \\phi_{n-3} + 124 \\phi_{n}\\right)}{90} + u_{n-1}$" ], "text/plain": [ " h⋅(29⋅\\phi_{n+1} + 24⋅\\phi_{n-1} + 4⋅\\phi_{n-2} - \\phi_{n-3} + 124⋅\\\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 90 \n", "\n", "phi_{n}) \n", "──────── + u_{n-1}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS-5: implicite, à 5 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(28 \\phi_{n+1} + 14 \\phi_{n-1} + 14 \\phi_{n-2} - 6 \\phi_{n-3} + \\phi_{n-4} + 129 \\phi_{n}\\right)}{90} + u_{n-1}$" ], "text/plain": [ " h⋅(28⋅\\phi_{n+1} + 14⋅\\phi_{n-1} + 14⋅\\phi_{n-2} - 6⋅\\phi_{n-3} + \\p\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 90 \n", "\n", "hi_{n-4} + 129⋅\\phi_{n}) \n", "──────────────────────── + u_{n-1}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS-6: implicite, à 6 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{h \\left(1139 \\phi_{n+1} + 33 \\phi_{n-1} + 1328 \\phi_{n-2} - 807 \\phi_{n-3} + 264 \\phi_{n-4} - 37 \\phi_{n-5} + 5640 \\phi_{n}\\right)}{3780} + u_{n-1}$" ], "text/plain": [ " h⋅(1139⋅\\phi_{n+1} + 33⋅\\phi_{n-1} + 1328⋅\\phi_{n-2} - 807⋅\\phi_{n-3\n", "u_{n+1} = ────────────────────────────────────────────────────────────────────\n", " 3780 \n", "\n", "} + 264⋅\\phi_{n-4} - 37⋅\\phi_{n-5} + 5640⋅\\phi_{n}) \n", "─────────────────────────────────────────────────── + u_{n-1}\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n" ] } ], "source": [ "Points = [(t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2), (t_nm3, phi_nm3),\n", " (t_nm4, phi_nm4), (t_nm5, phi_nm5)]\n", "for q in range(1, len(Points) + 1):\n", " pol = symb.interpolate(Points[:q], t)\n", " N = (symb.integrate(pol, (t, t_nm1, t_np1)).simplify())\n", " print(f\"N-{q}: explicite, à {q} pas\")\n", " display(symb.Eq(u_np1, u_nm1 + N))\n", " print(\"\\n\")\n", "print(\"\\n\")\n", "\n", "Points = [(t_np1, phi_np1), (t_n, phi_n), (t_nm1, phi_nm1), (t_nm2, phi_nm2),\n", " (t_nm3, phi_nm3), (t_nm4, phi_nm4), (t_nm5, phi_nm5)]\n", "for q in range(len(Points)):\n", " pol = symb.interpolate(Points[:q + 1], t)\n", " MS = (symb.integrate(pol, (t, t_nm1, t_np1)).simplify())\n", " print(f\"MS-{q}: implicite, à {q} pas\")\n", " display(symb.Eq(u_np1, u_nm1 + MS))\n", " print(\"\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Schémas BDF (Backward Differentiation Formulae)\n", "\n", "Soit $p$ le polynôme qui interpole la fonction inconnue $t\\mapsto y(t)$ en les points\n", "$$\n", "\\{t_{n+1},t_n,\\dots, t_{n-q+1}\\}\\text{ avec } q\\ge 1.\n", "$$ \n", "\n", "Au lieu d'évaluer l'EDO $\\varphi(t,y) = y'(t)$ en $t_{n+1}$, nous évaluons l'EDO approchée $\\varphi(t,y) = p'(t)$ en $t_{n+1}$ ce qui donne la relation\n", "$$\n", "\\varphi(t_{n+1},y_{n+1}) \\simeq p'(t_{n+1}).\n", "$$\n", "\n", "On peut construire différents schémas **implicites** selon les points d'interpolation utilisés pour approcher la fonction inconnue $t\\mapsto y(t)$. \n", "Cette solution approchée sera obtenue en construisant une suite récurrente comme suit:\n", "
\n", "\n", "$$\n", "\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1} \\text{ solution de l'équation }\\varphi(t_{n+1},u_{n+1}) = p'(t_{n+1})\n", "\\quad \\text{où $p(t)$ est un polynôme interpolant }y(t).\n", "\\end{cases}\n", "$$\n", "\n", "
\n", "\n", "\n", "Toutes ces méthodes peuvent s'écrire sous la forme\n", "
\n", "\n", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_1\\approx y_1,\\\\\n", "\\vdots\\\\\n", "u_{q-1}\\approx y_{q-1},\\\\\n", "\\displaystyle u_{n+1}=\\sum_{j=n-q+1}^{n}a_j u_j+hb_{-1}\\varphi(t_{n+1},u_{n+1}).\n", "\\end{cases}\n", "$$\n", "\n", "
\n", "\n", "On verra que les méthodes BDF-$q$ à $q$ pas sont d'ordre $q$ et sont zéro-stables ssi $q\\le6$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### BDF-1\n", "\n", "On a \n", "+ Points d'interpolation: $\\{(t_{n+1},y_{n+1}),(t_{n},y_{n})\\}$\n", "+ Polynôme: $p(t)=\\frac{y_{n+1}-y_{n}}{h}(t-t_{n})+y_{n}$\n", "+ Dérivée: $p'(t)=\\frac{y_{n+1}-y_{n}}{h}$\n", "+ Approximation: $\\varphi(t_{n+1},y_{n+1}) \\approx p'(t_{n+1}) = \\frac{y_{n+1}-y_{n}}{h}$\n", "et on obtient le schéma\n", "$$\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1} \\text{ solution de l'équation }\\varphi(t_{n+1},u_{n+1}) = \\frac{u_{n+1}-u_{n}}{h} & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "c'est-à-dire le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(BDF$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_n+h\\varphi(t_{n+1},u_{n+1}) & n=0,1,\\dots N-1.\n", "\\end{cases}$$\n", "\n", "
\n", "La méthode BDF$_1$ coïncide avec la méthode d'Euler implicite." ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle p{\\left(t \\right)} = \\frac{h u_{n} + t u_{n+1} - t u_{n} - t_{n} u_{n+1} + t_{n} u_{n}}{h}$" ], "text/plain": [ " h⋅u_{n} + t⋅u_{n+1} - t⋅u_{n} - tₙ⋅u_{n+1} + tₙ⋅u_{n}\n", "p(t) = ─────────────────────────────────────────────────────\n", " h " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left. \\frac{d}{d t} p{\\left(t \\right)} \\right|_{\\substack{ t=h + t_{n} }} = \\frac{u_{n+1} - u_{n}}{h}$" ], "text/plain": [ "⎛d ⎞│ u_{n+1} - u_{n}\n", "⎜──(p(t))⎟│ = ───────────────\n", "⎝dt ⎠│t=h + tₙ h " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\phi_{n+1} h + u_{n}$" ], "text/plain": [ "u_{n+1} = \\phi_{n+1}⋅h + u_{n}" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, u_np1), (t_n, u_n)], t).factor()\n", "# display(pol)\n", "display(symb.Eq(p(t),pol))\n", "dp = symb.diff(pol, t).subs(t, t_np1)\n", "display(symb.Eq(symb.diff(p(t),t).subs(t, t_np1),dp))\n", "rhs=symb.solve(dp - phi_np1, u_np1)[0]\n", "display(symb.Eq(u_np1,rhs))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### BDF-2\n", "\n", "On a \n", "+ Points d'interpolation: $\\{(t_{n+1},y_{n+1}),(t_{n},y_{n}),(t_{n-1},y_{n-1})\\}$\n", "+ Polynôme: \n", " $\\begin{aligned}[t]\n", " p(t)&=y_{n+1}\\frac{(t-t_n)(t-t_{n-1})}{(t_{n+1}-t_n)(t_{n+1}-t_{n-1})}+y_{n}\\frac{(t-t_{n+1})(t-t_{n-1})}{(t_{n}-t_{n+1})(t_{n}-t_{n-1})}+y_{n-1}\\frac{(t-t_{n+1})(t-t_{n})}{(t_{n-1}-t_{n+1})(t_{n-1}-t_{n})}\n", " \\\\\n", " &=\\frac{(t-t_{n})(t-t_{n-1})}{2h^2}y_{n+1}+\\frac{(t-t_{n+1})(t-t_{n-1})}{-h^2}y_{n}+\\frac{(t-t_{n+1})(t-t_{n})}{-2h^2}y_{n-1}\n", " \\end{aligned}$\n", "+ Dérivée: $p'(t)=\\frac{(t-t_{n})+(t-t_{n-1})}{2h^2}y_{n+1}+\\frac{(t-t_{n+1})+(t-t_{n-1})}{-h^2}y_{n}+\\frac{(t-t_{n+1})+(t-t_{n})}{-2h^2}y_{n-1}$\n", "+ Approximation: $\\varphi(t_{n+1},y_{n+1}) \\approx p'(t_{n+1}) = \\frac{3}{2h}y_{n+1}-\\frac{2}{h}y_{n}+\\frac{1}{2h}y_{n-1}$\n", "et on obtient le schéma\n", "$$\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1} \\text{ solution de l'équation }\\varphi(t_{n+1},u_{n+1}) = \\frac{3}{2h}u_{n+1}-\\frac{2}{h}u_{n}+\\frac{1}{2h}u_{n-1} & n=2,3,\\dots N-1\n", "\\end{cases}$$\n", "c'est-à-dire le schéma\n", "\n", "
\n", "\n", "$$\n", "\\text{(BDF$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{1},u_{1}) \\\\\n", "u_{n+1}=\\frac{4}{3}u_n-\\frac{1}{3}u_{n-1}+\\frac{2}{3}h\\varphi(t_{n+1},u_{n+1}) & n=1,2,3,\\dots N-1.\n", "\\end{cases}$$\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle p{\\left(t \\right)} = \\frac{h^{2} u_{n} + \\frac{h \\left(t u_{n+1} - t u_{n-1} - t_{n} u_{n+1} + t_{n} u_{n-1}\\right)}{2} + \\frac{t^{2} u_{n+1}}{2} + \\frac{t^{2} u_{n-1}}{2} - t^{2} u_{n} - t t_{n} u_{n+1} - t t_{n} u_{n-1} + 2 t t_{n} u_{n} + \\frac{t_{n}^{2} u_{n+1}}{2} + \\frac{t_{n}^{2} u_{n-1}}{2} - t_{n}^{2} u_{n}}{h^{2}}$" ], "text/plain": [ " 2 \n", " 2 h⋅(t⋅u_{n+1} - t⋅u_{n-1} - tₙ⋅u_{n+1} + tₙ⋅u_{n-1}) t ⋅u_{\n", " h ⋅u_{n} + ─────────────────────────────────────────────────── + ──────\n", " 2 2 \n", "p(t) = ───────────────────────────────────────────────────────────────────────\n", " \n", " \n", "\n", " 2 \n", "n+1} t ⋅u_{n-1} 2 tₙ\n", "──── + ────────── - t ⋅u_{n} - t⋅tₙ⋅u_{n+1} - t⋅tₙ⋅u_{n-1} + 2⋅t⋅tₙ⋅u_{n} + ──\n", " 2 \n", "──────────────────────────────────────────────────────────────────────────────\n", " 2 \n", " h \n", "\n", "2 2 \n", " ⋅u_{n+1} tₙ ⋅u_{n-1} 2 \n", "───────── + ─────────── - tₙ ⋅u_{n}\n", " 2 2 \n", "───────────────────────────────────\n", " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": 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ffB5KTaDYU5GxjDOe5UIB5VNA1CFUnrNN7vhnJ2ylgLnznzv+xJhoC7EnMblZZYx/NW4Z87CkzF8pMG3C3qDXkkgPhM3CfCZfZqxgsMRkyU9fgmbmsS+q1G5ry1EW5gv4ppCj1GWyKL4FLOLnri0XPHvzstGr0MbU1jpXIX+iZ73QkWWAe6a+zzuKLHduOqw6/072nexPsgC3MYFVCUjuWIaEItUru6sm5jIehkLbfeWBVSNPZd+rPLr3plLJFr/SRtx0clHAWWFCw/SnjkcN4snvGiZb/uvEZ4s/NV/Fd7IZptyscsa/MrfNyZPy/70sqzWOen0qxZ86I/NJfMmz8kO+FynzGfPTh3tSHvsiGHKrecyq8xV2VTnKyHwy35rbYjkaKpcTuk9m0U3r2nLB80uQjTu3t7ddVie5Fiw6pCimvTOtckdh4N2TT3XeUlx1zTtKD3QcrEB1jx1QP5U/GxNIRkAyReWE7DI72JLLZA9xRMUQkAwUseS4GKBOqAlkJJDz/6q4GaTn/fneZcYZs3XSqGGqBGz1vfaTspnzcMvRXxNYc/ida5hLlQvKc6iOudpQYX+lRPYqs3UaaRS635GsbtXhCE8DYmMCqxCQvKHMsssxqwbYkIjRRNxsLpdAs3rkchE45yZQDIFs/1faBFFAsWMvkoswdV6tzCYsbctRQphnFNUlygXFN1bHbGKGVglk9nVwuXGdifc6/iK/rKnfM3JnOTK7aPUqxXL3DO0eNTvMJSB5CjOzsTyyQgAF1zO1c8E6nAmYgAmYgAmYgAmYgAlMIJDtHVp16nmvkPfEHsk+k0UB4Jpdkzl/FnX8mX3de1dJ9/GPksDMK2HYAIplx2907L5zRnji6VVo5W5jAikJMICCXHJsjORycNl748knJmACJmACJmACJmACJmACSQhkU2iVOr6pxmwVCujvsi+DEqojG6HgjgKKeSi79ykg+UNJZSdjZnAf6xj863LPMIOLAmxzJAHxhPlUZmwZPvSpkyOfXL43MfD72OUXo3NgAiZgAiZgAiZgAiZQOIEsCq06+8zOvqnZMLPKu6/xzqMon/F7Jcx0faj99x1QZONPdPT5ITzPsjmSgMrEs4lHsrI3EzABEzABEzABEzABEzCB7RG4ypSkWIFFuf2x8xwU2J0Uquqo07uyN7gNGGYR95Ykd/yyuUOIr3Oruvxbn6PdTMAETMAETMAETMAETMAETMAEyiRwLaUy2Xd7FNcdMOhYKac6huWsXWWUGVd2hR1TYiui8oOSyswrW8GPmUNK8X/HAvteegIqu2SylT51jtEETMAETMAETMAETMAETKB0Aii0lRKaKSMorn2f2kHRjTdvYrnw0OxqpRQrnY1SjJIr21WGCT+2bPl/um8TERDDrO/QKv6cshXlxKcmYAImYAImYAImYAImYAKXSOA6c6ZRRhtFlGdJyXmuA8po/LkTlgsPvf/aen9W4dkgCv/dd2qZoc32TTnFfXZGLP0O7dmVqjNkAiZgAiZgAiZgAiZgApdDINc7tCiuzJjy/myjqMqNcxTZf+g8nmFFOeXzPn2mUVTrOD/XsavMEu6BbJ97X5x2MwETMAETMAETMAETMAETMAETKJxAzhnaaqmw+PDpnrCjMd+gfaLr7kwqm0bxGZ8+gwL8WmGY2d3p2P3+bAjD8zzjGGj4aAImYAImYAIFE1B7z8D4izoLYXA8/oZ9wblz0k3ABEzABFIRyKnQVkuF1SAxazo6c4of2Z3sfc7jzOka5Ze4Bo38VA2djq3lzYMBfMMETMAETMAETGDrBF6pXW8GqnXOvg/vZO9tPeFOnwmYgAmYwHoERhVaNR4sGf5BFoXxp7hhOSKJzJjG3549FOSVPNBwNY3XoQDRfWZxCW9jAiZgAiZgAiawUQLqRzDryoqsh7IfdD2mnD7X/Z9lw2A17Txue4PfcrcxARMwARO4UAJXY/lWo8HMKe+m0gC1Ppsjd3Yafi8blgE1UXFPF7i/aRwPnCgMux7zfuxefGNBa/+Ei3dNHgvieyZgAiZgAiZgAicgoLaaT/ZVX0DQ44OiOpQSBrjfDt20uwmYgAmYgAlAYFShxYMaHmZaMd2GB3eUzw/cDEb+2YX49/r6ha6rd1/D/QPHL3WfJUVTDP7nzOpOeYb9moAJmIAJmIAJpCPACrDWQHk3avUfvpe9idxp6/kUYOvVpOi+T03ABEzABC6QwOiS45pHNZLaaVS41esufywznrLUuH5MpTwzcvu17DeyQ5s/xf7ZbAr/3U2mGj8+MQETMAETMAET2A4BtdlDA+WDiVQYFGAGzFk1ZmMCJmACJmACDYFjFFoanu7sLBHgPktxJfCQqZXTg8os4eX3KH9Dz7K7CZiACZiACZjA6gR6B8SHUqG2ntVgvD/7QOfxjO1QELubgAmYgAlcEIGWQhs1Gsx4/imLIsuo6EvZne6jxLLkJ7wjy8YMbO7wRkcrlwJhYwImYAImYAImMEqgGihXv4EZ17uybAxFX2NvxZX8oMzy+T+U4F19zdErswBiYwImYAImsGsUWjUONDAop4yAVg2FjuH9lmqGVtccf9ORRuixjlUDo/MSzL9KSKTTaAImYAImYALnSkD9BgbEUV4xfJYn9DfofzALy14aldE9lFn2yUChDWEYVOfLBjYmYAImYAImUBGoFNq6gaExodGIRz05Z6fj7hIfFNnSNmX4u9JsYwImYAImYAImcDoCDJ5jnnX6G7ih7MaGb87ixrExCodSa2MCJmACJmACFYEwQ8uoKJ/h6X76hoan7z3ZIXdjNQETMAETMAETMIEhAtWAuPob3UFxZmBb/Q35+XQoErubgAmYgAmYQCBwVZ98pWNr4yc1JIyKstwnLDuuvA65Vzf9YwImYAImYAImYALDBBgQ7/Y36GtgfxwO5jsmYAImYAIm0E/gqlZQUV5biquuUXJ3ut9qeOREY9Ryr+PA2cYETMAETMAETMAE9gjUfYW9gXJ5ZAkxn+3rztruxWEHEzABEzABE+gSuIoc4ndncW7ek1Uj84UsjRCmceeCezqEezgVZ5SHd1H+ZqdfcTyXfS/L93GTG8WbJJ3JE+YITcAETMAETOAwgaGB8ucKWr3yVLejRfcpDmOwDxMwARMwgZQEmKG9UYTMwjYNiNxQUnmf5a0s5oncgsJ7V9fVudyqZck6FjuqWueBzw+F/FUZnvOjOGiQ4dOd1R6NTuHYxXHUpEzn6IN80wRMwARMwATyEGBAvK99pC8RlhvfS9Ee50m+YzUBEzABE9giges6UWyT/4MaEWYWP5OlYaHhYUt93GKFi+3yX8udEdWdjqV/f5Yl1EkUcrGoBgV0nBpfM5gA0wGTLJ0D8dvZBEzABEzABHISQHHliwpdQ7/iqdrOx7L+JE+Xjq9NwARMwARGCdy5vb0d9XAuN9VI/ke22TFR5yiRvLfDbHSYpf5R7lOV0QaRwqLkMxAQ3kdmoIBPIY3Gqfu/yhJuz8g9eTr3HmIHEzABEzABEzABEzABEzABEyiQQJihLTDpy5IsRZElxiibzHy+1LH5XIDOmZE+Ztb0Z/mNP3WEUvp5cNOR0egXss2H4nU+ySiOwXROisieTcAETMAETMAETMAETMAETODMCFysQhuVI+8Kt97pkRI596PtKMf/iOK+p/PF7+bW8e2lM3qOT03ABEzABEzABEzABEzABEzg4ghctEIrxRUFlE8FsOR4kVEc1YyujvHyYuJvvQ+k+32zvw/lHpYph3Q0s7+6lyydIXIfTcAETMAETMAETMAETMAETKB0Ahet0KrwmllPKY0sD2a58B+yfUpnX1k3SqduonSGXaF3igMFl/h+kWUX5UrR1XFv9ldug+/Q1g8dSifv7D6QZZMNnvdB9pHiaynRcrMxARMwARMwARMwARMwARMwgbMjcEkKbbwUOBQkS4Lf1Be86/qS8z6ls/Yzdog3g8IfCm54L/epzuOZW+5PMb3pVAQosO9leRe42lRKx1dcy9qYgAmYgAmYgAmYgAmYgAmYwFkTuDrr3EWZk6LXp1AyE8s375jpZIfjJUuPmeENCixPrt7LrRXMSlHGcabpTafi5nmPZCsFVtfM0i7Jw8zkOZgJmIAJmIAJmIAJmIAJmIAJrE/gYj7bsz7a458oRfTQkuPByBT2vSwzuDsd+WYwCi0bUb3VtZVbgbAxARMwARMwARMwARMwARM4TwIXM0O78eKbtROyFFZmheOZZ+JBueXdXSuzGy90J88ETMAETMAETMAETMAETGAZgf8DBaUTYPJEgUgAAAAASUVORK5CYII=\n", "text/latex": [ "$\\displaystyle \\left. \\frac{d}{d t} p{\\left(t \\right)} \\right|_{\\substack{ t=h + t_{n} }} = \\frac{\\frac{h \\left(u_{n+1} - u_{n-1}\\right)}{2} - t_{n} u_{n+1} - t_{n} u_{n-1} + 2 t_{n} u_{n} + u_{n+1} \\left(h + t_{n}\\right) + u_{n-1} \\left(h + t_{n}\\right) - 2 u_{n} \\left(h + t_{n}\\right)}{h^{2}}$" ], "text/plain": [ " h⋅(u_{n+1} - u_{n-1}) \n", " ───────────────────── - tₙ⋅u_{n+1} - tₙ⋅u_{n-1} + 2⋅tₙ⋅u\n", "⎛d ⎞│ 2 \n", "⎜──(p(t))⎟│ = ────────────────────────────────────────────────────────\n", "⎝dt ⎠│t=h + tₙ \n", " \n", "\n", " \n", "_{n} + u_{n+1}⋅(h + tₙ) + u_{n-1}⋅(h + tₙ) - 2⋅u_{n}⋅(h + tₙ)\n", " \n", "─────────────────────────────────────────────────────────────\n", " 2 \n", " h " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{2 \\phi_{n+1} h}{3} - \\frac{u_{n-1}}{3} + \\frac{4 u_{n}}{3}$" ], "text/plain": [ " 2⋅\\phi_{n+1}⋅h u_{n-1} 4⋅u_{n}\n", "u_{n+1} = ────────────── - ─────── + ───────\n", " 3 3 3 " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "pol = symb.interpolate([(t_np1, u_np1), (t_n, u_n), (t_nm1, u_nm1)],\n", " t).simplify()\n", "display(symb.Eq(p(t),pol))\n", "dp = symb.diff(pol, t).subs(t, t_np1)\n", "display(symb.Eq(symb.diff(p(t),t).subs(t, t_np1),dp))\n", "rhs=symb.solve(dp - phi_np1, u_np1)[0]\n", "display(symb.Eq(u_np1,rhs))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calcul systématique des coefficients des méthodes BDF" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "BDF-1: implicite, à 1 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\phi_{n+1} h + u_{n}$" ], "text/plain": [ "u_{n+1} = \\phi_{n+1}⋅h + u_{n}" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF-2: implicite, à 2 pas\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{2 \\phi_{n+1} h}{3} - \\frac{u_{n-1}}{3} + \\frac{4 u_{n}}{3}$" ], "text/plain": [ " 2⋅\\phi_{n+1}⋅h u_{n-1} 4⋅u_{n}\n", "u_{n+1} = ────────────── - ─────── + ───────\n", " 3 3 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF-3: implicite, à 3 pas\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{n+1} = \\frac{6 \\phi_{n+1} h}{11} - \\frac{9 u_{n-1}}{11} + \\frac{2 u_{n-2}}{11} + \\frac{18 u_{n}}{11}$" ], "text/plain": [ " 6⋅\\phi_{n+1}⋅h 9⋅u_{n-1} 2⋅u_{n-2} 18⋅u_{n}\n", "u_{n+1} = ────────────── - ───────── + ───────── + ────────\n", " 11 11 11 11 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF-4: implicite, à 4 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{12 \\phi_{n+1} h}{25} - \\frac{36 u_{n-1}}{25} + \\frac{16 u_{n-2}}{25} - \\frac{3 u_{n-3}}{25} + \\frac{48 u_{n}}{25}$" ], "text/plain": [ " 12⋅\\phi_{n+1}⋅h 36⋅u_{n-1} 16⋅u_{n-2} 3⋅u_{n-3} 48⋅u_{n}\n", "u_{n+1} = ─────────────── - ────────── + ────────── - ───────── + ────────\n", " 25 25 25 25 25 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF-5: implicite, à 5 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{60 \\phi_{n+1} h}{137} - \\frac{300 u_{n-1}}{137} + \\frac{200 u_{n-2}}{137} - \\frac{75 u_{n-3}}{137} + \\frac{12 u_{n-4}}{137} + \\frac{300 u_{n}}{137}$" ], "text/plain": [ " 60⋅\\phi_{n+1}⋅h 300⋅u_{n-1} 200⋅u_{n-2} 75⋅u_{n-3} 12⋅u_{n-4\n", "u_{n+1} = ─────────────── - ─────────── + ─────────── - ────────── + ─────────\n", " 137 137 137 137 137 \n", "\n", "} 300⋅u_{n}\n", "─ + ─────────\n", " 137 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF-6: implicite, à 6 pas\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle u_{n+1} = \\frac{20 \\phi_{n+1} h}{49} - \\frac{150 u_{n-1}}{49} + \\frac{400 u_{n-2}}{147} - \\frac{75 u_{n-3}}{49} + \\frac{24 u_{n-4}}{49} - \\frac{10 u_{n-5}}{147} + \\frac{120 u_{n}}{49}$" ], "text/plain": [ " 20⋅\\phi_{n+1}⋅h 150⋅u_{n-1} 400⋅u_{n-2} 75⋅u_{n-3} 24⋅u_{n-4\n", "u_{n+1} = ─────────────── - ─────────── + ─────────── - ────────── + ─────────\n", " 49 49 147 49 49 \n", "\n", "} 10⋅u_{n-5} 120⋅u_{n}\n", "─ - ────────── + ─────────\n", " 147 49 " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n" ] } ], "source": [ "Points = [(t_np1, u_np1), (t_n, u_n), (t_nm1, u_nm1), (t_nm2, u_nm2),\n", " (t_nm3, u_nm3), (t_nm4, u_nm4), (t_nm5, u_nm5)]\n", "for q in range(1, len(Points)):\n", " pol = symb.interpolate(Points[0:q + 1], t)\n", " dp = symb.diff(pol, t).subs(t, t_np1)\n", " sol = symb.solve(dp - phi_np1, u_np1)\n", " #display(sol)\n", " print(f\"BDF-{q}: implicite, à {q} pas\")\n", " display(symb.Eq(u_np1, sol[0]))\n", " print(\"\\n\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Schémas (multi-pas) de type *predictor-corrector*\n", "\n", "Lorsqu'on utilise une méthode implicite, pour calculer $u_{n+1}$ on doit résoudre une équation (en générale non-linéaire), par exemple avec une méthode de point fixe. \n", "Une approche différente qui permet de s'affranchir de cette étape est donnée par les méthodes **predictor-corrector**. Une méthode predictor-corrector est une méthode qui permet de calculer $u_{n+1}$ de façon explicite à partir d'une méthode implicite.\n", "\n", "
\n", "\n", "On considère une méthode implicite $u_{n+1}=\\displaystyle\\sum_{j=0}^p a_ju_{n-j} + h\\sum_{j=0}^p b_j\\varphi(t_{n-j},u_{n-j})+hb_{-1}\\varphi(t_{n+1},\\color{red}{u_{n+1}})$ pour $n=p,p+1,\\dots,N-1$.\n", "\n", "On modifie la méthode implicite comme suit:\n", "+ étape de **prédiction**: on calcule $\\tilde u_{n+1}$ une approximation de $u_{n+1}$ par une méthode explicite, typiquement une autre méthode multipas linéaire,\n", "+ étape de **correction**: on \"corrige\" la méthode implicite en remplacant $\\varphi(t_{n+1},\\color{red}{u_{n+1}})$ par $\\varphi(t_{n+1},\\color{red}{\\tilde u_{n+1}})$. \n", "\n", "Cela donne une récurrence du type\n", "$$\\begin{cases}\n", "\\tilde u_{n+1}=\\displaystyle\\sum_{j=0}^{\\tilde p} \\tilde a_ju_{n-j} + h\\sum_{j=0}^{\\tilde p} \\tilde b_j\\varphi(t_{n-j},u_{n-j})\n", "\\\\\n", "u_{n+1}=\\displaystyle\\sum_{j=0}^p a_ju_{n-j} + h\\sum_{j=0}^p b_j\\varphi(t_{n-j},u_{n-j})+hb_{-1}\\varphi(t_{n+1},\\color{red}{\\tilde u_{n+1}})\n", "\\end{cases}$$\n", " \n", " \n", "
\n", "\n", "\n", "### Remarques\n", "\n", "**Ordre**\n", "Parmi les méthodes linéaires à $q$ pas, les méthodes implicites sont, à nombre de pas égal, plus précises et en général plus stables (on verra ces propriétés lors du prochaine cours). Cependant, elles sont aussi plus coûteuses car le caractère implicite doit être résolu par une méthode itérative, de type point fixe par exemple. \n", "D’où l’idée, pour diminuer le coût, d’utiliser une méthode explicite pour calculer un bon prédicteur de la solution et de ne faire qu’une itération (ou quelques itérations) de la méthode implicite utilisée alors comme correcteur explicite.\n", "\n", "- Ces méthodes héritent de l’ordre de précision du correcteur. (La contribution à l’erreur du prédicteur est d’un ordre inférieur à celle du correcteur). Pour avoir une méthode d’ordre $\\omega$ on pourra prendre un prédicteur d’ordre $\\omega-1$ et un correcteur d’ordre $\\omega$ (typiquement deux méthodes de Adam ayant le même nombre de pas). \n", " Schématiquement, si on note\n", " - $\\omega_{[C]}$ l'ordre du corrector \n", " - $\\omega_{[P]}$ l'ordre du predictor \n", " \n", " alors l'ordre du predictor-corrector est $\\omega_{[PC]}=\\min\\{\\omega_{[C]},\\omega_{[P]}+1\\}$\n", "\n", "\n", "**Zéro-stabilité**\n", "On démontre que la zéro-stabilité du prédicteur n’influe pas sur la zéro-stabilité de la méthode prédicteur-correcteur. On peut même utiliser un prédicteur instable.\n", "\n", "**A-stabilité**\n", "Elles sont soumises à une condition de A-stabilité qui est typiquement celle du prédicteur. Elles ne sont donc pas adaptées à la résolution des problèmes de Cauchy sur des intervalles non bornés ou des problèmes stiff.\n", "\n", "\n", "\n", "\n", "\n", "### Exemple : schéma de Heun\n", "Considérons les deux schémas PC suivants:\n", "+ *predictor*: méthode d'Euler explicite (ou AB$_1$) $\\tilde u_{n+1}=u_n+h\\varphi(t_n,u_n)$\n", "+ *corrector*: méthode de Crank-Nicolson (ou AM$_1$) $u_{n+1}=u_n+\\frac{h}{2}\\left(\\varphi(t_n,u_n)+\\varphi(t_{n+1},u_{n+1})\\right)$\n", "\n", "On retrouve la méthode de Heun\n", "
\n", "\n", "$$\\text{(H)}\\quad\n", "\\begin{cases}\n", "\t\tu_0\t = y_0 \\\\\n", "\\tilde \tu_{n+1} = u_n + h\\varphi(t_n,u_n)\t\\\\\n", "\t\tu_{n+1} = u_n + \\frac{h}{2} \\left(\\varphi(t_{n},u_{n})+\\varphi(t_{n+1},\\tilde u_{n+1})\\right) & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple : AB2-AM2\n", "Des méthodes de type predictor-corrector sont souvent construites en utilisant une prédiction d'Adam-Bashforth suivie d'une correction d'Adam-Moulton. \n", "Par exemple, si on considère les deux étapes suivantes\n", "+ *predictor*: méthode AB$_2$ $\\tilde u_{n+1}=u_{n}+ \\frac{h}{2} \\left(3\\varphi(t_{n},u_{n})-\\varphi(t_{n-1},u_{n-1})\\right)$\n", "+ *corrector*: méthode AM$_2$ $u_{n+1}=u_{n}+\\frac{h}{12}\\left(5\\varphi(t_{n+1},u_{n+1})+8\\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1})\\right)$\n", "\n", "On obtient la méthode AB$_2$-AM$_2$:\n", "
\n", "\n", "$$\\text{(AB$_2$-AM$_2$)}\\quad\n", "\\begin{cases}\n", "\t\tu_0 = y_0 \\\\\n", "\t\tu_1 = u_0 +h\\varphi(t_0,u_0),\\\\\n", "\\tilde \tu_{n+1} = u_n +\\frac{3}{2}h\\left( \\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1}) \\right) \\\\\n", "\t\tu_{n+1} = u_{n}+\\frac{h}{12}\\left(5\\varphi(t_{n+1},\\tilde u_{n+1})+8\\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1})\\right) & n=2,3,\\dots N-1\n", "\\end{cases}$$ \n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exemple : AB2-BDF2\n", "Des méthodes de type predictor-corrector sont souvent construites en utilisant une prédiction d'Adam-Bashforth suivie d'une correction BDF. \n", "Par exemple, si on considère les deux étapes suivantes\n", "+ *predictor*: méthode AB$_2$ $\\tilde u_{n+1}=u_{n}+ \\frac{h}{2} \\left(3\\varphi(t_{n},u_{n})-\\varphi(t_{n-1},u_{n-1})\\right)$\n", "+ *corrector*: méthode BDF$_2$ $u_{n+1}=\\frac{4}{3}u_n-\\frac{1}{3}u_{n-1}+\\frac{2}{3}h\\varphi(t_{n+1},u_{n+1})$\n", "\n", "On obtient la méthode AB$_2$-BDF$_2$:\n", "
\n", "\n", "$$\\text{(AB$_2$-BDF$_2$)}\\quad\n", "\\begin{cases}\n", "\t\tu_0 = y_0 \\\\\n", "\t\tu_1 = u_0 +h\\varphi(t_0,u_0),\\\\\n", "\\tilde \tu_{n+1} = u_n +\\frac{3}{2}h\\left( \\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1}) \\right)\\\\\n", "\t\tu_{n+1}=\\frac{4}{3}u_n-\\frac{1}{3}u_{n-1}+\\frac{2}{3}h\\varphi(t_{n+1},\\tilde u_{n+1}) & n=1,2,3,\\dots N-1.\n", "\\end{cases}$$ \n", "\n", "
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