{ "cells": [ { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "execution_count": 46, "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": 47, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python version 3.8.5 (default, Jan 27 2021, 15:41:15) \n", "[GCC 9.3.0]\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", "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", "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", "Les schémas qu'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", "$$\n", "u_{n+1} \n", "= \n", "\\sum_{j=0}^p a_ju_{n-j}\n", "+\n", "h\\sum_{j=0}^p b_j\\varphi(t_{n-j},u_{n-j})\n", "+\n", "hb_{-1}\\varphi(t_{n+1},u_{n+1}),\n", "\\qquad\n", "n=p,p+1,\\dots,N-1\n", "$$\n", "où les $\\{a_k\\}$ et $\\{b_k\\}$ sont des coefficients donnés et $p\\ge0$ un entier.\n", "
\n", "\n", "\n", "**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", "$$\n", "u_{n+1} \n", "= \n", "a_0u_{n}\n", "+\n", "hb_0\\varphi(t_{n},u_{n})\n", "+\n", "hb_{-1}\\varphi(t_{n+1},u_{n+1}),\n", "$$\n", "avec\n", "\n", "| \t | | Type | $a_0$ | $b_0$ | $b_{-1}$ \t|\n", "|-----\t|--------------------------------------------------------------------------------- |---------- |------\t|------\t|----------\t|\n", "| EE \t| $u_{n+1}=u_n+h\\varphi(t_n,u_n)$ | Explicite | 1 \t| 1 \t| 0 \t|\n", "| EI \t| $u_{n+1}=u_n+h\\varphi(t_{n+1},u_{n+1})$ | Implicite | 1 \t| 0 \t| 1 \t|\n", "| CN \t| $u_{n+1}=u_n+\\frac{h}{2}\\left(\\varphi(t_n,u_n)+\\varphi(t_{n+1},u_{n+1})\\right)$ | Implicite | 2 \t| $\\frac{1}{2}$| $\\frac{1}{2}$|\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Remarques pour l'implémentation**\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" ] }, { "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": 48, "metadata": {}, "outputs": [], "source": [ "import sympy as symb\n", "symb.init_printing()\n", "\n", "symb.var('h,t,t_n')\n", "# Pour Adam et N-MS\n", "symb.var('phi_np1,phi_n,phi_nm1,phi_nm2,phi_nm3,phi_nm4,phi_nm5')\n", "# Pour BDF\n", "symb.var(' y_np1, y_n, y_nm1, y_nm2, y_nm3, y_nm4, y_nm5')\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" ] }, { "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 différentes 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", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1}=u_n+\\displaystyle\\int_{t_n}^{t_{n+1}} \\tilde f(t) \\mathrm{d}t \n", "\\quad \\text{où $\\tilde f(t)$ est un polynôme interpolant }\\varphi(t,y(t)) \n", "\\end{cases}\n", "$$\n", "
\n", "\n", "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 différentes schémas selon les points d'interpolation choisis. \n", "\n", "Toutes ces méthodes peuvent s'écrire sous la forme\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", "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", "$$\n", "\\text{(AB$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_{n}+h \\varphi(t_{n},u_{n})& n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "
\n", "La méthode AB$_1$ coïncide avec la méthode d'Euler progressive. " ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\phi_{n}$" ], "text/plain": [ "φₙ" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle h \\phi_{n}$" ], "text/plain": [ "h⋅φₙ" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n)], t)\n", "display(p)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(AB$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n}+ \\dfrac{h}{2} \\big(3\\varphi(t_{n},u_{n})-\\varphi(t_{n-1},u_{n-1})\\big)& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AB$_1$. \n", "
" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{h \\left(3 \\phi_{n} - \\phi_{nm1}\\right)}{2}$" ], "text/plain": [ "h⋅(3⋅φₙ - φₙₘ₁)\n", "───────────────\n", " 2 " ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n),(t_nm1,phi_nm1)], t)\n", "# display(p)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(AB$_3$)}\\quad\n", "\\begin{cases}\n", "u_0 = y(t_0) = y_0,\\\\\n", "u_1 = u_0 + h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_2 = u_1 + \\dfrac{h}{2} \\big(3\\varphi(t_1,u_1)-\\varphi(t_0,u_0)\\big)\\approx y(t_2)\\\\\n", "u_{n+1} = u_{n} + \\dfrac{h}{12}\\big(23\\varphi(t_n,u_n)-16\\varphi(t_{n-1},u_{n-1})+5\\varphi(t_{n-2},u_{n-2})\\big) & n=2,3,\\dots N-1\n", "\\end{cases}$$\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", "
" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{h \\left(23 \\phi_{n} - 16 \\phi_{nm1} + 5 \\phi_{nm2}\\right)}{12}$" ], "text/plain": [ "h⋅(23⋅φₙ - 16⋅φₙₘ₁ + 5⋅φₙₘ₂)\n", "────────────────────────────\n", " 12 " ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n),(t_nm1,phi_nm1),(t_nm2,phi_nm2)], t)\n", "# display(p)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(AB$_4$)}\\quad\n", "\\begin{cases}\n", "u_0 = y(t_0) = y_0,\\\\\n", "u_1 = u_0 + h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_2 = u_1 + \\dfrac{h}{2} \\big(3\\varphi(t_1,u_1)-\\varphi(t_0,u_0)\\big)\\approx y(t_2)\\\\\n", "u_{3} = u_{2} + \\dfrac{h}{12}\\big(23\\varphi(t_2,u_2)-16\\varphi(t_{1},u_{1})+5\\varphi(t_{0},u_{0})\\big)\\approx y(t_3) \\\\\n", "u_{n+1} = u_{n} + \\dfrac{h}{24}\\big(55\\varphi(t_n,u_n)-59\\varphi(t_{n-1},u_{n-1})+37\\varphi(t_{n-2},u_{n-2})-9\\varphi(t_{n-3},u_{n-3})\\big) & n=3,4,\\dots N-1\n", "\\end{cases}$$\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", "
" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{h \\left(55 \\phi_{n} - 59 \\phi_{nm1} + 37 \\phi_{nm2} - 9 \\phi_{nm3}\\right)}{24}$" ], "text/plain": [ "h⋅(55⋅φₙ - 59⋅φₙₘ₁ + 37⋅φₙₘ₂ - 9⋅φₙₘ₃)\n", "──────────────────────────────────────\n", " 24 " ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n),(t_nm1,phi_nm1),(t_nm2,phi_nm2),(t_nm3,phi_nm3)], t)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(AM$_0$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_{n}+h \\varphi(t_{n+1},u_{n+1})& n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "
\n", "La méthode AM$_0$ coïncide avec la méthode d'Euler regressive. " ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle h \\phi_{np1}$" ], "text/plain": [ "h⋅φₙₚ₁" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,phi_np1)], t)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{t-t_n}{t_{n+1}-t_n}\n", " +\\varphi(t_{n},y_{n})\\frac{t-t_{n+1}}{t_n-t_{n+1}}$\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", "$$\n", "\\text{(AM$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_{n}+ \\dfrac{h}{2} \\big(\\varphi(t_{n},u_{n})+\\varphi(t_{n+1},u_{n+1})\\big)& n=0,1,\\dots N-1\n", "\\end{cases}$$\n", "
\n", "La méthode AM$_1$ coïncide avec la méthode de Crank-Nicolson. " ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{h \\left(\\phi_{n} + \\phi_{np1}\\right)}{2}$" ], "text/plain": [ "h⋅(φₙ + φₙₚ₁)\n", "─────────────\n", " 2 " ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,phi_np1),(t_n,phi_n)], t)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{(t-t_n)(t-t_{n-1})}{(t_{n+1}-t_n)(t_{n+1}-t_{n-1})}\n", " +\\varphi(t_{n},y_{n})\\frac{(t-t_{n+1})(t-t_{n-1})}{(t_n-t_{n+1})(t_n-t_{n-1})}\n", " +\\varphi(t_{n-1},y_{n-1})\\frac{(t-t_{n+1})(t-t_{n})}{(t_{n-1}-t_{n+1})(t_{n+1}-t_{n})}$\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", "$$\n", "\\text{(AM$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_{0}+ \\dfrac{h}{2} \\big(\\varphi(t_{1},u_{1})+\\varphi(t_{0},u_{0})\\big)\\\\\n", "u_{n+1}=u_{n}+\\dfrac{h}{12}\\big(5\\varphi(t_{n+1},u_{n+1})+8\\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1})\\big) & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AM$_1$. \n", "
" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{h \\left(8 \\phi_{n} - \\phi_{nm1} + 5 \\phi_{np1}\\right)}{12}$" ], "text/plain": [ "h⋅(8⋅φₙ - φₙₘ₁ + 5⋅φₙₚ₁)\n", "────────────────────────\n", " 12 " ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,phi_np1),(t_n,phi_n),(t_nm1,phi_nm1)], t)\n", "symb.integrate(p,(t,t_n,t_np1)).simplify()" ] }, { "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": 56, "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": [ "$\\displaystyle u_{np1} = h \\phi_{n} + u_{n}$" ], "text/plain": [ "uₙₚ₁ = h⋅φₙ + uₙ" ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(3 \\phi_{n} - \\phi_{nm1}\\right)}{2} + u_{n}$" ], "text/plain": [ " h⋅(3⋅φₙ - φₙₘ₁) \n", "uₙₚ₁ = ─────────────── + uₙ\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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(23 \\phi_{n} - 16 \\phi_{nm1} + 5 \\phi_{nm2}\\right)}{12} + u_{n}$" ], "text/plain": [ " h⋅(23⋅φₙ - 16⋅φₙₘ₁ + 5⋅φₙₘ₂) \n", "uₙₚ₁ = ──────────────────────────── + uₙ\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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\n", "text/latex": [ "$\\displaystyle u_{np1} = \\frac{h \\left(55 \\phi_{n} - 59 \\phi_{nm1} + 37 \\phi_{nm2} - 9 \\phi_{nm3}\\right)}{24} + u_{n}$" ], "text/plain": [ " h⋅(55⋅φₙ - 59⋅φₙₘ₁ + 37⋅φₙₘ₂ - 9⋅φₙₘ₃) \n", "uₙₚ₁ = ────────────────────────────────────── + uₙ\n", " 24 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(1901 \\phi_{n} - 2774 \\phi_{nm1} + 2616 \\phi_{nm2} - 1274 \\phi_{nm3} + 251 \\phi_{nm4}\\right)}{720} + u_{n}$" ], "text/plain": [ " h⋅(1901⋅φₙ - 2774⋅φₙₘ₁ + 2616⋅φₙₘ₂ - 1274⋅φₙₘ₃ + 251⋅φₙₘ₄) \n", "uₙₚ₁ = ────────────────────────────────────────────────────────── + uₙ\n", " 720 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(4277 \\phi_{n} - 7923 \\phi_{nm1} + 9982 \\phi_{nm2} - 7298 \\phi_{nm3} + 2877 \\phi_{nm4} - 475 \\phi_{nm5}\\right)}{1440} + u_{n}$" ], "text/plain": [ " h⋅(4277⋅φₙ - 7923⋅φₙₘ₁ + 9982⋅φₙₘ₂ - 7298⋅φₙₘ₃ + 2877⋅φₙₘ₄ - 475⋅φₙₘ₅) \n", "uₙₚ₁ = ────────────────────────────────────────────────────────────────────── \n", " 1440 \n", "\n", " \n", "+ uₙ\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": "iVBORw0KGgoAAAANSUhEUgAAAKgAAAAVCAYAAADSH6UfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAErklEQVRoBe2a4VEUMRTHV4YCGDpQOgDsQDtAqEDsQMZvfGOgA6UCkQ60A5UOsAMZOsDfL2zWXO52j9vb1eXYzIRskpfk5Z9/3nu7x7O7u7tiTNMIHB8fb9D6hbxLvqG+NS01v4Vxh+RP8yVHiVkIrM1qHNuKAlLdkl+DxS/ytyUwaUXsJdZbqaEjQecf5zYiX+eL1Ur8ru0ZYAeX8hX5cCiqjQRtOAkPq+xuZUHL8VcNSwyxy9DGPIg0ErT5GIKLh2i3zWK1vduMbUXu2hmfWMe6+wVEb4xm3XjpJ/WJoJ66LwtvKdseFMMfZdKCfmPfe5SbZPHR5b+jzdh0KtFu/wFZ1/6aOkUYd8TzSuPH/jrnUSAoAH5gcgH0IM7JFUFpk7h7lG8oB5XQ6SMKSaJF0tVD9oKMYEs20yn1QEhKL+speQoP+tSnoHxXlj6fkV9Q9+JL2JnEVn4FUuc8WgcwD+F7CY4u7SYDyrZBxlHoHoiQ6dtVNRJfz5GTSvJOJGQkrp+jIjkdH9y748k+zyT2xESPtML+euGRFlTwIgH3qZ9kGAl03paJLFdlfQ/ctXWdO8vN1tnocDETbOLEHsRlrFgio5cRp+fWy6S1PIoVyluyMp0n1lkYP8bUeZ9NFaR/1uVv8j698EgLKnAqJHhuNHXvuibbegv0WdcDjwfnWkNJ6pQTUTzMnzMlJeJFxDLri1X3mVvi2Ne6bIsf42YRsKDdMO8F5dkiSiHfC4/WEyWMqbwFYaGyXWUL2qKFLZu7K8q5vZlhrUVmZkydFWiapskKhHHM60WRiPn3Tw/VD/g5Hsr+JIdUjr/O6hJ+Km6NMm3LUpdW+LVdc864TnmUElSQc+B1czGO0o0FKwEoPuuKPZQfZME3bdFX3cxM7l6CN1seTkpgY1urMl2r1QT1gww3CubPPYf7Dh6GPp99w9cqiltq/cUjHRtePJENFrkc+9/xQ8c+Uqc8Wks0nHA/gKhL2iVH0ko+rUc8iPg2u0ubb6q6hH1KFSwoUzktnYfp4Z6QHTvkVF3MTEn3FN27eETM3tJeXUyeX8Y+yvzlKcXFl6ZVwy9iwtYCD5biUWpBjaPOAVTQ/IbnQt5yP7G8p4wHs8nzBVkLYkhQxazUBf+GbErlLpGLit/SF0gcpIb5x33MukRidMBeXpGrFyCeQ9hA6UXUtfuBXhydJ/cWKS6riF+nPKoICqASZ1aMNNGGXCAa5UvkI2kL6ro1LazzWI9yWqM0lvNGBRnlhpjQXZ2nEu21Lw706Wn8CuHlE4f04lZz0R5xWUn82F+nPFqrkFv8IY+zJLKWtUBJSRiTYUIajx1QD3JRYMVK957ut257Q8VPgv1LA9KIw3odek3tEFArsUEZ41PFXegNbb6NRysR5KiHDZfjPMChfOtElc6T8efE56l8hRKHQeKHbg+5XPmWWtUfgkMrgqKNxMsPwfhL63idHJCklcT+VEoRfpPe4bm6oaWSvmAo6/e3EANT1rpTJxpwMn6fl0b87hGai8OzPv+jHpJJWuOx6oVi3smN/X8RGPErirW/cPTypFX83svMT2PSJ49fbwTl9uuqNeH+Jr3xNPjU3S5H/O6x/AMUPmJ2i0j17AAAAABJRU5ErkJggg==\n", "text/latex": [ "$\\displaystyle u_{np1} = h \\phi_{np1} + u_{n}$" ], "text/plain": [ "uₙₚ₁ = h⋅φₙₚ₁ + uₙ" ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(\\phi_{n} + \\phi_{np1}\\right)}{2} + u_{n}$" ], "text/plain": [ " h⋅(φₙ + φₙₚ₁) \n", "uₙₚ₁ = ───────────── + uₙ\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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(8 \\phi_{n} - \\phi_{nm1} + 5 \\phi_{np1}\\right)}{12} + u_{n}$" ], "text/plain": [ " h⋅(8⋅φₙ - φₙₘ₁ + 5⋅φₙₚ₁) \n", "uₙₚ₁ = ──────────────────────── + uₙ\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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(19 \\phi_{n} - 5 \\phi_{nm1} + \\phi_{nm2} + 9 \\phi_{np1}\\right)}{24} + u_{n}$" ], "text/plain": [ " h⋅(19⋅φₙ - 5⋅φₙₘ₁ + φₙₘ₂ + 9⋅φₙₚ₁) \n", "uₙₚ₁ = ────────────────────────────────── + uₙ\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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(646 \\phi_{n} - 264 \\phi_{nm1} + 106 \\phi_{nm2} - 19 \\phi_{nm3} + 251 \\phi_{np1}\\right)}{720} + u_{n}$" ], "text/plain": [ " h⋅(646⋅φₙ - 264⋅φₙₘ₁ + 106⋅φₙₘ₂ - 19⋅φₙₘ₃ + 251⋅φₙₚ₁) \n", "uₙₚ₁ = ───────────────────────────────────────────────────── + uₙ\n", " 720 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(1427 \\phi_{n} - 798 \\phi_{nm1} + 482 \\phi_{nm2} - 173 \\phi_{nm3} + 27 \\phi_{nm4} + 475 \\phi_{np1}\\right)}{1440} + u_{n}$" ], "text/plain": [ " h⋅(1427⋅φₙ - 798⋅φₙₘ₁ + 482⋅φₙₘ₂ - 173⋅φₙₘ₃ + 27⋅φₙₘ₄ + 475⋅φₙₚ₁) \n", "uₙₚ₁ = ───────────────────────────────────────────────────────────────── + uₙ\n", " 1440 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(65112 \\phi_{n} - 46461 \\phi_{nm1} + 37504 \\phi_{nm2} - 20211 \\phi_{nm3} + 6312 \\phi_{nm4} - 863 \\phi_{nm5} + 19087 \\phi_{np1}\\right)}{60480} + u_{n}$" ], "text/plain": [ " h⋅(65112⋅φₙ - 46461⋅φₙₘ₁ + 37504⋅φₙₘ₂ - 20211⋅φₙₘ₃ + 6312⋅φₙₘ₄ - 863⋅φₙ\n", "uₙₚ₁ = ───────────────────────────────────────────────────────────────────────\n", " 60480 \n", "\n", "ₘ₅ + 19087⋅φₙₚ₁) \n", "──────────────── + uₙ\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n" ] } ], "source": [ "import sympy as symb\n", "symb.init_printing()\n", "\n", "symb.var('h,t,t_n')\n", "symb.var('phi_np1,phi_n,phi_nm1,phi_nm2,phi_nm3,phi_nm4,phi_nm5')\n", "symb.var(' u_np1, u_n, u_nm1, u_nm2, u_nm3, u_nm4, u_nm5')\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", "Points=[(t_n,phi_n),(t_nm1,phi_nm1),(t_nm2,phi_nm2),(t_nm3,phi_nm3),(t_nm4,phi_nm4),(t_nm5,phi_nm5)]\n", "for q in range(1,len(Points)+1):\n", " p=symb.interpolate(Points[:q], t)\n", " AB=(symb.integrate(p,(t,t_n,t_np1)).simplify())\n", " print(\"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),(t_nm3,phi_nm3),(t_nm4,phi_nm4),(t_nm5,phi_nm5)]\n", "for q in range(len(Points)):\n", " p=symb.interpolate(Points[0:q+1], t)\n", " AM=(symb.integrate(p,(t,t_n,t_np1)).simplify())\n", " print(\"AM-\",q, \": implicite, à\",q,\"pas, d'ordre\", q+1)\n", " display(symb.Eq(u_np1,u_n+AM))\n", " print(\"\\n\")\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 différentes 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", "$$\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_1=?,\\\\\n", "u_{n+1}=u_n+\\displaystyle\\int_{t_{n-1}}^{t_{n+1}} \\tilde f(t) \\mathrm{d}t \n", "\\quad \\text{où $\\tilde f(t)$ est un polynôme interpolant }\\varphi(t,y(t)) \n", "\\end{cases}\n", "$$\n", "
\n", "\n", "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 différentes schémas selon les points d'interpolation choisis. \n", "Toutes ces méthodes peuvent s'écrire à nouveau sous la forme\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", "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", "$$\n", "\\text{(N$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n},u_{n})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler explicite. \n", "
\n", "\n", "La méthode N$_1$ coïncide avec la **méthode du point milieu** (appelée aussi **Saute-mouton** ou **Leapfrog**). " ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 2 h \\phi_{n}$" ], "text/plain": [ "2⋅h⋅φₙ" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n)], t).simplify()\n", "symb.integrate(p,(t,t_nm1,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(N$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n},u_{n})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler explicite. \n", "
\n", "On retrouve la méthode N$_1$." ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 2 h \\phi_{n}$" ], "text/plain": [ "2⋅h⋅φₙ" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n),(t_nm1,phi_nm1)], t).simplify()\n", "symb.integrate(p,(t,t_nm1,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(N$_3$)}\\quad\n", "\\begin{cases}\n", "u_0 = y(t_0) = y_0,\\\\\n", "u_1 = u_0 + h\\varphi(t_{0},u_{0})\\approx y(t_1)\\\\\n", "u_2 = u_{0} + 2h\\varphi(t_{1},u_{1})\\approx y(t_2)\\\\\n", "u_{n+1} = u_{n-1} + \\dfrac{h}{3}\\big(7\\varphi(t_n,u_n)-2\\varphi(t_{n-1},u_{n-1})+\\varphi(t_{n-2},u_{n-2})\\big) & n=2,3,\\dots N-1\n", "\\end{cases}$$\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", "
" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{h \\left(7 \\phi_{n} - 2 \\phi_{nm1} + \\phi_{nm2}\\right)}{3}$" ], "text/plain": [ "h⋅(7⋅φₙ - 2⋅φₙₘ₁ + φₙₘ₂)\n", "────────────────────────\n", " 3 " ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_n,phi_n),(t_nm1,phi_nm1),(t_nm2,phi_nm2)], t).simplify()\n", "symb.integrate(p,(t,t_nm1,t_np1)).simplify()" ] }, { "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", "$$\n", "\\text{(MS$_0$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{1},u_{1})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n+1},u_{n+1})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler implicite.\n", "
" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 2 h \\phi_{np1}$" ], "text/plain": [ "2⋅h⋅φₙₚ₁" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,phi_np1)], t).simplify()\n", "symb.integrate(p,(t,t_nm1,t_np1)).simplify()" ] }, { "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: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{t-t_n}{t_{n+1}-t_n}\n", " +\\varphi(t_{n},y_{n})\\frac{t-t_{n+1}}{t_n-t_{n+1}}$\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", "$$\n", "\\text{(MS$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{1},u_{1})\\approx y(t_1)\\\\\n", "u_{n+1}=u_{n-1}+2h \\varphi(t_{n},u_{n})& n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction d'Euler implicite. \n", "
\n", "On retrouve la méthode N$_1$: le schéma est devenu explicite." ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle 2 h \\phi_{n}$" ], "text/plain": [ "2⋅h⋅φₙ" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,phi_np1),(t_n,phi_n)], t).simplify()\n", "symb.integrate(p,(t,t_nm1,t_np1)).simplify()" ] }, { "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: $p(t)=\\varphi(t_{n+1},y_{n+1})\\frac{(t-t_n)(t-t_{n-1})}{(t_{n+1}-t_n)(t_{n+1}-t_{n-1})}\n", " +\\varphi(t_{n},y_{n})\\frac{(t-t_{n+1})(t-t_{n-1})}{(t_n-t_{n+1})(t_n-t_{n-1})}\n", " +\\varphi(t_{n-1},y_{n-1})\\frac{(t-t_{n+1})(t-t_{n})}{(t_{n-1}-t_{n+1})(t_{n+1}-t_{n})}$\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", "$$\n", "\\text{(MS$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_{0}+ \\dfrac{h}{2} \\big(\\varphi(t_{1},u_{1})+\\varphi(t_{0},u_{0})\\big)\\\\\n", "u_{n+1}=u_{n-1}+\\dfrac{h}{3}\\big( \\varphi(t_{n+1},u_{n+1})+4\\varphi(t_n,u_n)+\\varphi(t_{n-1},u_{n-1}\\big) & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "où $u_{1}$ est une approximation de $y(t_1)$ obtenue en utilisant une prédiction AM$_1$ (=Crank-Nicolson). \n", "
" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\frac{h \\left(4 \\phi_{n} + \\phi_{nm1} + \\phi_{np1}\\right)}{3}$" ], "text/plain": [ "h⋅(4⋅φₙ + φₙₘ₁ + φₙₚ₁)\n", "──────────────────────\n", " 3 " ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,phi_np1),(t_n,phi_n),(t_nm1,phi_nm1)], t).simplify()\n", "symb.integrate(p,(t,t_nm1,t_np1)).simplify()" ] }, { "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": 63, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "N- 1 : explicite, à 1 pas\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle u_{np1} = 2 h \\phi_{n} + u_{nm1}$" ], "text/plain": [ "uₙₚ₁ = 2⋅h⋅φₙ + uₙₘ₁" ] }, "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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"text/latex": [ "$\\displaystyle u_{np1} = 2 h \\phi_{n} + u_{nm1}$" ], "text/plain": [ "uₙₚ₁ = 2⋅h⋅φₙ + uₙₘ₁" ] }, "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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"text/latex": [ "$\\displaystyle u_{np1} = \\frac{h \\left(7 \\phi_{n} - 2 \\phi_{nm1} + \\phi_{nm2}\\right)}{3} + u_{nm1}$" ], "text/plain": [ " h⋅(7⋅φₙ - 2⋅φₙₘ₁ + φₙₘ₂) \n", "uₙₚ₁ = ──────────────────────── + uₙₘ₁\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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(8 \\phi_{n} - 5 \\phi_{nm1} + 4 \\phi_{nm2} - \\phi_{nm3}\\right)}{3} + u_{nm1}$" ], "text/plain": [ " h⋅(8⋅φₙ - 5⋅φₙₘ₁ + 4⋅φₙₘ₂ - φₙₘ₃) \n", "uₙₚ₁ = ───────────────────────────────── + uₙₘ₁\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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cENa82ZJHTPUsHtCG+SAvn+ZofdttqAzRTnLBcqgMxxkBI2AEjIARMAK3G4El6Ep9LVDSodi5y/8CBwXrZScDdsa6YR2S/b0RsLT7hpK3CCO+/HXnRbSEmTACRsAIGAEj0EZAc/TYR4/7TvYuyEnpS5tDQyd7N6ortWv/n0/1YINuS4di5y7co5NLJMpVc99OYbyCTNhsn0RQGeGbN3I5El6MET+DHWcxjJoRI2AEjIARMAJGoIWA5vCS8raJugafVtnpxUbR36iu1Kpc5hFfRR3qLKPhHhqaaahADA8KV0zckMrPPTyOT/l203vZpDXy1ekG0A6dyILhmJePFR581Ks8+jTzWFTRGdLMiwkcaASMgBEwAkbACBgBITC3rpTrVHyb8Iksbw2jm1E270eM6k+5cscuXTcBiljYtVNmKHT5F7t5gxZLQUEDlsv/w/EhQpREdvww0PB1fRQ/wmGcsOq/WRFt0SivRpEsEjjQCBgBI2AEjIARMALTITCnrlTSqfiv7fASqFz0qheywT9UpbtZZOs7MsqEc9wHsknhQzlDc+S8GiWPeJS15hhXfgq+lMXkdPl3asgDcGyMgBEwAkbACBgBI7AmBObUlUo61csMHDbFWuVnca3HfOeOvzF6JWUt/GVRzICjV3bi2BJ8LbvRc8hY7sMUFsM5muV/+lDecjp2//L/xUQpDDTQ2RgBI2AEjIARMAJGYCUIzKUrdXWqoCtJp0obbMCDntX85eQQXo1yF5Wy0lZfKSwVkmuU0IVjW+X1ecYQu385M08THZnYGAEjYASmQEBjDndwk+HkoPjhddGlPyDnv0RZCYerJCmhXSNgBIxAHwIaP9icKulFpTCyQSGr0ZW6dPi5xhaMyuXEkxdBOAnNdaxI0XbyY9l2zIAvFnIut6tR/qSw5q3XRKes0m4ezKGN5sreQEmOMgJGwAgMI6BxhrHog6g+yOVbmdzFZTD9W8+tKyDy8xLYO7nprjBjEX9W3qJTmI0RMAKniQD6yFFOD+O4Uq0riT7XqR6Kz/zUE2UvfbmEMW7QnA3G9kcyEP7aiWbVzK4cA2yKgxmY5R8u5IRV8hd6boDVM3nBKLRopeFYWG54SUNhNkbACBiBIQT4V52LfMzQM2/Fs+plXOJqyEZ+XgpjoE3jE2FX0d/QQWtjBIzAaSIgeU8K0jEqWKsrlegIyzfC4PuR+OeaXBMuf1GHCv8tO1cNVSgDJoNnw8hcZTlfI2AEbicCGl/44DjKXOvfbeRnochA+ImeGYdYBeO2jk/k57SBt9QCnVwbI2AEjMCqEbg7M/fsxr2buQxnbwSMwC1FQIrZeaz6ZQEC7tRhuPeLYTwq0aW3z4i3MQJGwAisHoHZlDsNuqya2S5kGzENwKsHzBUwAkZgOQhobElXPHiBoms+jQFc96gZg0p5dPO03wgYASOweATO5uJQgylHsT6OnQtg52sEjEBCgDt0pV03Xt7CoNglxS0pgyGi81OjAHaS2GsEjIARWB4Cs+3cLa+q5sgIGIETReAZ9dKCslHw9IxilxS5dOw6Vv200zdG53gjYASMwKIRmG3nbtG1NnNGwAicDAJS5HhJ4p4qxPfqUOpQ0rjri+VlCZS70l07BQeTdvXSHb0UbtcIGAEjsEoErNytstnMtBEwAjkCKHjyt779pLD0sWL+0gcFkCSlo9cUVrvDRz42RsAIGIHFIuBj2cU2jRkzAkbgQATYxfstKn5kxXeieMmra9LO3TG/f9XlwX4jYASMwGQIeOduMiidkREwAjeBgJQ3jl75kPG9pMjJZTfuK9n823d8yy7t5umxMdDwnTx2/2yMgBEwAqtHwDt3q29CV8AI3HoE2I3r3qlDkeOvyP5M6Oj5Z+jkogwGo2eUwK9lw0sZ16H+NQJGwAisG4FZ/6Fi3dCYeyNgBNaCgJS0tCOX7s/9orCtY9aozEHLLh0vUPD/jS8V3iiB8tsYASNgBFaNwP8BAAz97DQaPbgAAAAASUVORK5CYII=\n", "text/latex": [ "$\\displaystyle u_{np1} = \\frac{h \\left(269 \\phi_{n} - 266 \\phi_{nm1} + 294 \\phi_{nm2} - 146 \\phi_{nm3} + 29 \\phi_{nm4}\\right)}{90} + u_{nm1}$" ], "text/plain": [ " h⋅(269⋅φₙ - 266⋅φₙₘ₁ + 294⋅φₙₘ₂ - 146⋅φₙₘ₃ + 29⋅φₙₘ₄) \n", "uₙₚ₁ = ───────────────────────────────────────────────────── + uₙₘ₁\n", " 90 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(297 \\phi_{n} - 406 \\phi_{nm1} + 574 \\phi_{nm2} - 426 \\phi_{nm3} + 169 \\phi_{nm4} - 28 \\phi_{nm5}\\right)}{90} + u_{nm1}$" ], "text/plain": [ " h⋅(297⋅φₙ - 406⋅φₙₘ₁ + 574⋅φₙₘ₂ - 426⋅φₙₘ₃ + 169⋅φₙₘ₄ - 28⋅φₙₘ₅) \n", "uₙₚ₁ = ──────────────────────────────────────────────────────────────── + uₙₘ₁\n", " 90 " ] }, "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": [ "$\\displaystyle u_{np1} = 2 h \\phi_{np1} + u_{nm1}$" ], "text/plain": [ "uₙₚ₁ = 2⋅h⋅φₙₚ₁ + uₙₘ₁" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "MS- 1 : implicite, à 1 pas\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle u_{np1} = 2 h \\phi_{n} + u_{nm1}$" ], "text/plain": [ "uₙₚ₁ = 2⋅h⋅φₙ + uₙₘ₁" ] }, "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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"text/latex": [ "$\\displaystyle u_{np1} = \\frac{h \\left(4 \\phi_{n} + \\phi_{nm1} + \\phi_{np1}\\right)}{3} + u_{nm1}$" ], "text/plain": [ " h⋅(4⋅φₙ + φₙₘ₁ + φₙₚ₁) \n", "uₙₚ₁ = ────────────────────── + uₙₘ₁\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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"text/latex": [ "$\\displaystyle u_{np1} = \\frac{h \\left(4 \\phi_{n} + \\phi_{nm1} + \\phi_{np1}\\right)}{3} + u_{nm1}$" ], "text/plain": [ " h⋅(4⋅φₙ + φₙₘ₁ + φₙₚ₁) \n", "uₙₚ₁ = ────────────────────── + uₙₘ₁\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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(124 \\phi_{n} + 24 \\phi_{nm1} + 4 \\phi_{nm2} - \\phi_{nm3} + 29 \\phi_{np1}\\right)}{90} + u_{nm1}$" ], "text/plain": [ " h⋅(124⋅φₙ + 24⋅φₙₘ₁ + 4⋅φₙₘ₂ - φₙₘ₃ + 29⋅φₙₚ₁) \n", "uₙₚ₁ = ────────────────────────────────────────────── + uₙₘ₁\n", " 90 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(129 \\phi_{n} + 14 \\phi_{nm1} + 14 \\phi_{nm2} - 6 \\phi_{nm3} + \\phi_{nm4} + 28 \\phi_{np1}\\right)}{90} + u_{nm1}$" ], "text/plain": [ " h⋅(129⋅φₙ + 14⋅φₙₘ₁ + 14⋅φₙₘ₂ - 6⋅φₙₘ₃ + φₙₘ₄ + 28⋅φₙₚ₁) \n", "uₙₚ₁ = ──────────────────────────────────────────────────────── + uₙₘ₁\n", " 90 " ] }, "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": [ "$\\displaystyle u_{np1} = \\frac{h \\left(5640 \\phi_{n} + 33 \\phi_{nm1} + 1328 \\phi_{nm2} - 807 \\phi_{nm3} + 264 \\phi_{nm4} - 37 \\phi_{nm5} + 1139 \\phi_{np1}\\right)}{3780} + u_{nm1}$" ], "text/plain": [ " h⋅(5640⋅φₙ + 33⋅φₙₘ₁ + 1328⋅φₙₘ₂ - 807⋅φₙₘ₃ + 264⋅φₙₘ₄ - 37⋅φₙₘ₅ + 1139\n", "uₙₚ₁ = ───────────────────────────────────────────────────────────────────────\n", " 3780 \n", "\n", "⋅φₙₚ₁) \n", "────── + uₙₘ₁\n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n" ] } ], "source": [ "import sympy as symb\n", "symb.init_printing()\n", "symb.var('h,t,t_n')\n", "symb.var('phi_np1,phi_n,phi_nm1,phi_nm2,phi_nm3,phi_nm4,phi_nm5')\n", "symb.var(' u_np1, u_n, u_nm1, u_nm2, u_nm3, u_nm4, u_nm5')\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", "Points=[(t_n,phi_n),(t_nm1,phi_nm1),(t_nm2,phi_nm2),(t_nm3,phi_nm3),(t_nm4,phi_nm4),(t_nm5,phi_nm5)]\n", "for q in range(1,len(Points)+1):\n", " p=symb.interpolate(Points[:q], t)\n", " N=(symb.integrate(p,(t,t_nm1,t_np1)).simplify())\n", " print(\"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),(t_nm3,phi_nm3),(t_nm4,phi_nm4),(t_nm5,phi_nm5)]\n", "for q in range(len(Points)):\n", " p=symb.interpolate(Points[:q+1], t)\n", " MS=(symb.integrate(p,(t,t_nm1,t_np1)).simplify())\n", " print(\"MS-\",q, \": implicite, à\",q,\"pas\")\n", " display(symb.Eq(u_np1,u_nm1+MS))\n", " print(\"\\n\")\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", "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", "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", "$$\n", "\\begin{cases}\n", "u_0=y_0,\\\\\n", "u_{n+1} \\text{ solution de l'équation }\\varphi(t_{n+1},u_{n+1}) = p'(t_{n+1})\n", "\\quad \\text{où $p(t)$ est un polynôme interpolant }y(t).\n", "\\end{cases}\n", "$$\n", "
\n", "\n", "\n", "Toutes ces méthodes peuvent s'écrire sous la forme\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", "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", "$$\n", "\\text{(BDF$_1$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_{n+1}=u_n+h\\varphi(t_{n+1},u_{n+1}) & n=0,1,\\dots N-1.\n", "\\end{cases}$$\n", "
\n", "La méthode BDF$_1$ coïncide avec la méthode d'Euler implicite." ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p(t)\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\frac{- h y_{n} + t y_{n} - t y_{np1} - t_{n} y_{n} + t_{n} y_{np1}}{h}$" ], "text/plain": [ "-(-h⋅yₙ + t⋅yₙ - t⋅yₙₚ₁ - tₙ⋅yₙ + tₙ⋅yₙₚ₁) \n", "───────────────────────────────────────────\n", " h " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "p'(t)\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\frac{y_{n} - y_{np1}}{h}$" ], "text/plain": [ "-(yₙ - yₙₚ₁) \n", "─────────────\n", " h " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "y_np1=\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle h \\phi_{np1} + y_{n}$" ], "text/plain": [ "h⋅φₙₚ₁ + yₙ" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,y_np1),(t_n,y_n)], t).factor()\n", "print(\"p(t)\")\n", "display(p)\n", "dp=symb.diff(p,t).subs(t,t_np1)\n", "print(\"p'(t)\")\n", "display(dp)\n", "print(\"y_np1=\")\n", "symb.solve(dp-phi_np1,y_np1)[0]" ] }, { "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: $\\begin{aligned}[t]\n", " p(t)&=y_{n+1}\\frac{(t-t_n)(t-t_{n-1})}{(t_{n+1}-t_n)(t_{n+1}-t_{n-1})}\n", " +y_{n}\\frac{(t-t_{n+1})(t-t_{n-1})}{(t_{n}-t_{n+1})(t_{n}-t_{n-1})}\n", " +y_{n-1}\\frac{(t-t_{n+1})(t-t_{n})}{(t_{n-1}-t_{n+1})(t_{n-1}-t_{n})}\n", " \\\\\n", " &=\\frac{(t-t_{n})(t-t_{n-1})}{2h^2}y_{n+1}\n", " +\\frac{(t-t_{n+1})(t-t_{n-1})}{-h^2}y_{n}\n", " +\\frac{(t-t_{n+1})(t-t_{n})}{-2h^2}y_{n-1}\n", " \\end{aligned}$\n", "+ Dérivée: $p'(t)=\\frac{(t-t_{n})+(t-t_{n-1})}{2h^2}y_{n+1}\n", " +\\frac{(t-t_{n+1})+(t-t_{n-1})}{-h^2}y_{n}\n", " +\\frac{(t-t_{n+1})+(t-t_{n})}{-2h^2}y_{n-1}$\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", "$$\n", "\\text{(BDF$_2$)}\\quad\n", "\\begin{cases}\n", "u_0=y(t_0)=y_0,\\\\\n", "u_1=u_0+h\\varphi(t_{1},u_{1}) \\\\\n", "u_{n+1}=\\frac{4}{3}u_n-\\frac{1}{3}u_{n-1}+\\frac{2}{3}h\\varphi(t_{n+1},u_{n+1}) & n=1,2,3,\\dots N-1.\n", "\\end{cases}$$\n", "
" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p(t)\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle \\frac{h^{2} y_{n} + \\frac{h \\left(- t y_{nm1} + t y_{np1} + t_{n} y_{nm1} - t_{n} y_{np1}\\right)}{2} - t^{2} y_{n} + \\frac{t^{2} y_{nm1}}{2} + \\frac{t^{2} y_{np1}}{2} + 2 t t_{n} y_{n} - t t_{n} y_{nm1} - t t_{n} y_{np1} - t_{n}^{2} y_{n} + \\frac{t_{n}^{2} y_{nm1}}{2} + \\frac{t_{n}^{2} y_{np1}}{2}}{h^{2}}$" ], "text/plain": [ " 2 2 \n", " 2 h⋅(-t⋅yₙₘ₁ + t⋅yₙₚ₁ + tₙ⋅yₙₘ₁ - tₙ⋅yₙₚ₁) 2 t ⋅yₙₘ₁ t ⋅yₙₚ₁ \n", "h ⋅yₙ + ──────────────────────────────────────── - t ⋅yₙ + ─────── + ─────── +\n", " 2 2 2 \n", "──────────────────────────────────────────────────────────────────────────────\n", " 2 \n", " h \n", "\n", " 2 2 \n", " 2 tₙ ⋅yₙₘ₁ tₙ ⋅yₙₚ₁\n", " 2⋅t⋅tₙ⋅yₙ - t⋅tₙ⋅yₙₘ₁ - t⋅tₙ⋅yₙₚ₁ - tₙ ⋅yₙ + ──────── + ────────\n", " 2 2 \n", "─────────────────────────────────────────────────────────────────\n", " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "p'(t)\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle \\frac{\\frac{h \\left(- y_{nm1} + y_{np1}\\right)}{2} + 2 t_{n} y_{n} - t_{n} y_{nm1} - t_{n} y_{np1} - 2 y_{n} \\left(h + t_{n}\\right) + y_{nm1} \\left(h + t_{n}\\right) + y_{np1} \\left(h + t_{n}\\right)}{h^{2}}$" ], "text/plain": [ "h⋅(-yₙₘ₁ + yₙₚ₁) \n", "──────────────── + 2⋅tₙ⋅yₙ - tₙ⋅yₙₘ₁ - tₙ⋅yₙₚ₁ - 2⋅yₙ⋅(h + tₙ) + yₙₘ₁⋅(h + tₙ)\n", " 2 \n", "──────────────────────────────────────────────────────────────────────────────\n", " 2 \n", " h \n", "\n", " \n", " + yₙₚ₁⋅(h + tₙ)\n", " \n", "────────────────\n", " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "y_np1=\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{2 h \\phi_{np1}}{3} + \\frac{4 y_{n}}{3} - \\frac{y_{nm1}}{3}$" ], "text/plain": [ "2⋅h⋅φₙₚ₁ 4⋅yₙ yₙₘ₁\n", "──────── + ──── - ────\n", " 3 3 3 " ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p=symb.interpolate([(t_np1,y_np1),(t_n,y_n),(t_nm1,y_nm1)], t).simplify()\n", "print(\"p(t)\")\n", "display(p)\n", "dp=symb.diff(p,t).subs(t,t_np1)\n", "print(\"p'(t)\")\n", "display(dp)\n", "print(\"y_np1=\")\n", "symb.solve(dp-phi_np1,y_np1)[0]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Calcul systématique des coefficients des méthodes BDF" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "BDF- 1\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ h \\phi_{np1} + u_{n}\\right]$" ], "text/plain": [ "[h⋅φₙₚ₁ + uₙ]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF- 2\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ \\frac{2 h \\phi_{np1}}{3} + \\frac{4 u_{n}}{3} - \\frac{u_{nm1}}{3}\\right]$" ], "text/plain": [ "⎡2⋅h⋅φₙₚ₁ 4⋅uₙ uₙₘ₁⎤\n", "⎢──────── + ──── - ────⎥\n", "⎣ 3 3 3 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF- 3\n" ] }, { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left[ \\frac{6 h \\phi_{np1}}{11} + \\frac{18 u_{n}}{11} - \\frac{9 u_{nm1}}{11} + \\frac{2 u_{nm2}}{11}\\right]$" ], "text/plain": [ "⎡6⋅h⋅φₙₚ₁ 18⋅uₙ 9⋅uₙₘ₁ 2⋅uₙₘ₂⎤\n", "⎢──────── + ───── - ────── + ──────⎥\n", "⎣ 11 11 11 11 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF- 4\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle \\left[ \\frac{12 h \\phi_{np1}}{25} + \\frac{48 u_{n}}{25} - \\frac{36 u_{nm1}}{25} + \\frac{16 u_{nm2}}{25} - \\frac{3 u_{nm3}}{25}\\right]$" ], "text/plain": [ "⎡12⋅h⋅φₙₚ₁ 48⋅uₙ 36⋅uₙₘ₁ 16⋅uₙₘ₂ 3⋅uₙₘ₃⎤\n", "⎢───────── + ───── - ─────── + ─────── - ──────⎥\n", "⎣ 25 25 25 25 25 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF- 5\n" ] }, { "data": { "image/png": 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\n", "text/latex": [ "$\\displaystyle \\left[ \\frac{60 h \\phi_{np1}}{137} + \\frac{300 u_{n}}{137} - \\frac{300 u_{nm1}}{137} + \\frac{200 u_{nm2}}{137} - \\frac{75 u_{nm3}}{137} + \\frac{12 u_{nm4}}{137}\\right]$" ], "text/plain": [ "⎡60⋅h⋅φₙₚ₁ 300⋅uₙ 300⋅uₙₘ₁ 200⋅uₙₘ₂ 75⋅uₙₘ₃ 12⋅uₙₘ₄⎤\n", "⎢───────── + ────── - ──────── + ──────── - ─────── + ───────⎥\n", "⎣ 137 137 137 137 137 137 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n", "BDF- 6\n" ] }, { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ \\frac{20 h \\phi_{np1}}{49} + \\frac{120 u_{n}}{49} - \\frac{150 u_{nm1}}{49} + \\frac{400 u_{nm2}}{147} - \\frac{75 u_{nm3}}{49} + \\frac{24 u_{nm4}}{49} - \\frac{10 u_{nm5}}{147}\\right]$" ], "text/plain": [ "⎡20⋅h⋅φₙₚ₁ 120⋅uₙ 150⋅uₙₘ₁ 400⋅uₙₘ₂ 75⋅uₙₘ₃ 24⋅uₙₘ₄ 10⋅uₙₘ₅⎤\n", "⎢───────── + ────── - ──────── + ──────── - ─────── + ─────── - ───────⎥\n", "⎣ 49 49 49 147 49 49 147 ⎦" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\n", "\n" ] } ], "source": [ "import sympy as symb\n", "symb.init_printing()\n", "symb.var('h,t,t_n')\n", "symb.var('phi_np1')\n", "symb.var(' u_np1, u_n, u_nm1, u_nm2, u_nm3, u_nm4, u_nm5')\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", "Points=[(t_np1,u_np1),(t_n,u_n),(t_nm1,u_nm1),(t_nm2,u_nm2),(t_nm3,u_nm3),(t_nm4,u_nm4),(t_nm5,u_nm5)]\n", "for q in range(1,len(Points)):\n", " print(\"BDF-\",q)\n", " p=symb.interpolate(Points[0:q+1], t)\n", " dp=symb.diff(p,t).subs(t,t_np1)\n", " sol=symb.solve(dp-phi_np1,u_np1)\n", " display(sol)\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 comme suit.\n", "\n", "
\n", "On considère une méthode implicite $u_{n+1}=u_n+hG(u_n,u_{n-1}\\dots,u_{n-q+1})+hb_{-1}\\varphi(t_{n+1},u_{n+1})$:\n", "\n", "
\n", "\n", "\n", "**Remarques** \n", "- On verra que, parmi les méthodes linéaires à $q$ pas, les méthodes implicites sont, à nombre de pas égal, plus précises, elles sont en général aussi plus stables. Mais elles sont plus coûteuses car le caractère implicite doit être résolu par une méthode itérative, de type point fixe par exemple. 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", "- 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", "- 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", "- Cependant, étant explicites, 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", "On retrouve la méthode de Heun\n", "
\n", "$$\n", "\\text{(H)}\\quad\n", "\\begin{cases}\n", "\t\tu_0\t = y_0 \\\\\n", "\\tilde \tu_{n+1} = u_n + h\\varphi(t_n,u_n)\t& n=1,2,\\dots N-1\\\\\n", "\t\tu_{n+1} = u_n + \\frac{h}{2} \\left(\\varphi(t_{n},u_{n})+\\varphi(t_{n+1},\\tilde u_{n+1})\\right) & n=1,2,\\dots N-1\n", "\\end{cases}$$\n", "
\n", "\n", "\n", "### 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", "on obtient la méthode AB$_2$-AM$_2$:\n", "
\n", "$$\n", "\\text{(AB$_2$-AM$_2$)}\\quad\n", "\\begin{cases}\n", "\t\tu_0 = y_0 \\\\\n", "\t\tu_1 = u_0 +h\\varphi(t_0,u_0),\\\\\n", "\\tilde \tu_{n+1} = u_n +\\frac{3}{2}h\\left( \\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1}) \\right)& n=2,3,\\dots N-1\\\\\n", "\t\tu_{n+1} = u_{n}+\\frac{h}{12}\\left(5\\varphi(t_{n+1},\\tilde u_{n+1})+8\\varphi(t_n,u_n)-\\varphi(t_{n-1},u_{n-1})\\right) & n=2,3,\\dots N-1\n", "\\end{cases}$$ \n", "
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