TP 4 - Étude de la convergence

%reset -f
%matplotlib inline
%autosave 300

import sys #only needed to determine Python version number
print('Python version ' + sys.version)

import numpy as np
import matplotlib.pyplot as plt
plt.rcParams.update({<span class="st">'font.size'</span>: <span class="dv">12</span>})

from scipy.optimize import fsolve

import sympy as sym
sym.init_printing()

Autosaving every 300 seconds
Python version 3.12.3 (main, Feb  4 2025, 14:48:35) [GCC 13.3.0]

1 Exercice : Estimation de l’Ordre de Convergence des Schémas Numériques

🎯 Objectif : compléter l’implémentation des schémas numériques indiqués et estimer l’ordre de convergence de chaque schéma sur le problème de Cauchy

\( \begin{cases} y'(t) = y(t), &\forall t \in I = [0,1], \\ y(0) = 1. \end{cases} \)

dont la solution exacte est \(y(t) = e^{t}\).

🛠️ Résultats attendus :

1. Méthodes multi-pas explicites à \(q\) pas (\(\text{AB}_q\) et \(\text{N}_q\)) : convergence à l’ordre \(q\).

2. Méthodes multi-pas implicites à \(q\) pas (\(\text{AM}_q\) et \(\text{MS}_q\)) : convergence à l’ordre …

   • \(q + 1\) si \(q\) est impair,
   • \(q + 2\) si \(q\) est pair.

3. Méthodes multi-pas implicites à \(q\) pas (\(\text{BDF}_q\)) : convergence à l’ordre \(q\).

4. Méthodes predictor-corrector : convergence à l’ordre \(\min(\omega_E + 1, \omega_I)\).

   • \(\omega_E\) : ordre du schéma predictor (explicite),
   • \(\omega_I\) : ordre du schéma corrector (implicite).

5. Méthodes de Runge-Kutta à \(s\) étages : convergence à l’ordre …

   • \(\le s\) si elles sont explicites,
   • \(\le 2s\) si elles sont implicites.

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🔎 Remarque :

1. Puisque la fonction \(\varphi(t, y) = y\) est linéaire, toute méthode implicite peut être rendue explicite par un calcul élémentaire.
   Cela consiste à expliciter directement, pour chaque schéma, l’expression de \(u_{n+1}\). Cependant, nous pouvons utiliser le module SciPy pour résoudre ces équations sans modifier l’implémentation des schémas. Attention : l’utilisation de fsolve peut réduire l’ordre de convergence global au même ordre que celui de fsolve lui-même. Dans notre exemple ce n’est pas le cas car la fonction \(\varphi\) est linéaire, mais il est important de le garder à l’esprit.

2. Initialisation des schémas multi-step : les schémas multi-step nécessitent l’initialisation de plusieurs pas pour pouvoir démarrer. Plutôt que d’utiliser un schéma d’ordre inférieur pour l’initialisation (ce qui dégraderait l’ordre de précision global), nous utiliserons la solution exacte.

3. Comme d’habitude, on notera \(\varphi_j = \varphi(t_j, u_j)\).

Rappels de la démarche pour le calcul de l’orde de convergence.

• On écrit les schémas numériques :
  • les nœuds d’intégration \([t_0,t_1,\dots,t_{N}]\) sont contenus dans le vecteur tt (qui change en fonction de h)
  • les valeurs \([u_0,u_1,\dots,u_{N}]\) pour chaque méthode sont contenues dans le vecteur uu.
• Pour chaque schéma, on calcule la solution approchée avec différentes valeurs de \(h_k=1/N_k\). On sauvegarde les valeurs de \(h_k\) dans le vecteur H.
• Pour chaque valeur de \(h_k\), on calcule le maximum de la valeur absolue de l’erreur et on sauvegarde toutes ces erreurs dans le vecteur err_schema de sort que err_schema[k] contient \(e_k=\max_{i=0,\dots,N_k}|y(t_i)-u_{i}|\).
• Pour afficher l’ordre de convergence on utilise une échelle logarithmique, i.e. on représente \(\ln(h)\) sur l’axe des abscisses et \(\ln(\text{err})\) sur l’axe des ordonnées.
  En effet, si \(\text{err}=Ch^p\) alors \(\ln(\text{err})=\ln(C)+p\ln(h)\).
  En échelle logarithmique, \(p\) représente donc la pente de la ligne droite \(\ln(\text{err})\).

2 Schémas explicites

2.1 Schéma de Adam-Bashforth à 1 pas = schéma d’Euler progressif

\( \begin{cases} u_0=y_0,\\ u_{n+1}=u_n+h \varphi_n & n=0,1,2,\dots N-1 \end{cases} \)

def EE(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = phi(tt[n],uu[n])
        uu[n+1] = uu[n]+h*k1
    return uu

2.2 Schéma de Adam-Bashforth à 2 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{n+1}=u_n+\frac{h}{2}\Bigl(3 \varphi_n - \varphi_{n-1} \Bigr)& n=1,2,3,4,5,\dots N-1 \end{cases} \)

def AB2(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 2
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        uu[n+1] = uu[n] + (3*k1-k2)*h/2
    return uu

2.3 Schéma de Adam-Bashforth à 3 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{2}=y_2,\\ u_{n+1}=u_n+\frac{h}{12}\Bigl(23 \varphi_n -16 \varphi_{n-1} +5 \varphi_{n-2} \Bigr)& n=2,3,4,5,\dots N-1 \end{cases} \)

def AB3(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 3
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        uu[n+1] = uu[n] + (23*k1-16*k2+5*k3)*h/12
    return uu

2.4 Schéma de Adam-Bashforth à 4 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{2}=y_2,\\ u_{3}=y_3,\\ u_{n+1}=u_n+\frac{h}{24}\Bigl(55 \varphi_n -59 \varphi_{n-1} +37 \varphi_{n-2} -9 \varphi_{n-3} \Bigr)& n=3,4,5,\dots N-1 \end{cases} \)

def AB4(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 4
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        k4 = phi( tt[n-3], uu[n-3] )
        uu[n+1] = uu[n] + (55*k1-59*k2+37*k3-9*k4)*h/24
    return uu

2.5 Schéma de Adam-Bashforth à 5 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{2}=y_2,\\ u_{3}=y_3,\\ u_{4}=y_4,\\ u_{n+1}=u_n+\frac{h}{720}\Bigl(1901 \varphi_n -2774 \varphi_{n-1} +2616 \varphi_{n-2} -1274 \varphi_{n-3} +251 \varphi_{n-4} \Bigr)& n=4,5,\dots N-1 \end{cases} \)

def AB5(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 5
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        k4 = phi( tt[n-3], uu[n-3] )
        k5 = phi( tt[n-4], uu[n-4] )
        uu[n+1] = uu[n] + (1901*k1-2774*k2+2616*k3-1274*k4+251*k5)*h/720
    return uu

2.6 Schéma de Nylström à 2 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{n+1}=u_{n-1}+2h \varphi_{n} & n=1,2,3,4,5,\dots N-1 \end{cases} \)

def N2(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 2
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        uu[n+1] = uu[n-1] + 2*h*k1
    return uu

2.7 Schéma de Nylström à 3 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{2}=y_2,\\ u_{n+1}=u_{n-1}+\frac{h}{3}\Bigl(7 \varphi_{n} -2 \varphi_{n-1} + \varphi_{n-2} \Bigr)& n=2,3,4,5,\dots N-1 \end{cases} \)

def N3(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 3
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        uu[n+1] = uu[n-1] + (7*k1-2*k2+k3)*h/3
    return uu

2.8 Schéma de Nylström à 4 pas

\( \begin{cases} u_0=y_0,\\ u_{1}=y_1,\\ u_{2}=y_2,\\ u_{3}=y_3,\\ u_{n+1}=u_{n-1}+\frac{h}{3}\Bigl(8 \varphi_{n} -5 \varphi_{n-1} +4 \varphi_{n-2} - \varphi_{n-3} \Bigr)& n=3,4,5,\dots N-1 \end{cases} \)

def N4(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 4
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        k4 = phi( tt[n-3], uu[n-3] )
        uu[n+1] = uu[n-1] + (8*k1-5*k2+4*k3-k4)*h/3
    return uu

2.9 Schéma d’Euler modifié (RK explicite à 2 pas)

\( \begin{cases} u_0=y_0,\\ \tilde u = u_n+\frac{h}{2} \varphi_n ,\\ u_{n+1}=u_n+h\varphi\left(t_n+\frac{h}{2},\tilde u\right)& n=0,1,2,\dots N-1 \end{cases} \)

def EM(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        uu[n+1] = uu[n]+h*phi(tt[n]+h/2,uu[n]+k1*h/2)
    return uu

2.10 Schéma de Runge-Kutta RK4-1

\( \begin{cases} u_0 = y_0 \\ K_1 = \varphi\left(t_n,u_n\right)\\ K_2 = \varphi\left(t_n+\frac{h}{2},u_n+\frac{h}{2} K_1)\right)\\ K_3 = \varphi\left(t_n+\frac{h}{2},u_n+\frac{h}{2}K_2\right)\\ K_4 = \varphi\left(t_{n+1},u_n+h K_3\right)\\ u_{n+1} = u_n + \frac{h}{6}\left(K_1+2K_2+2K_3+K_4\right) & n=0,1,\dots N-1 \end{cases} \)

def RK4(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n]+h/2 , uu[n]+h*k1/2 )
        k3 = phi( tt[n]+h/2 , uu[n]+h*k2/2 )
        k4 = phi( tt[n+1]    , uu[n]+h*k3 )
        uu[n+1] = uu[n] + (k1+2*k2+2*k3+k4)*h/6
    return uu

2.11 Schéma de Runge-Kutta RK6_5 (Butcher à 6 étages d’ordre 5)

\(\begin{array}{c|cccccc} 0 & 0 & 0 & 0 & 0 &0&0 \\ \frac{1}{4} & \frac{1}{4} & 0&0&0&0&0\\ \frac{1}{4} & \frac{1}{8} &\frac{1}{8}&0&0&0&0\\ \frac{1}{2} & 0 &-\frac{1}{2}&1&0&0&0\\ \frac{3}{4} & \frac{3}{16} &0&0&\frac{9}{16}&0&0\\ 1 & \frac{-3}{7} &\frac{2}{7}&\frac{12}{7}&\frac{-12}{7}&\frac{8}{7}&0\\ \hline & \frac{7}{90} & 0&\frac{32}{90} & \frac{12}{90} & \frac{32}{90} & \frac{7}{90} \end{array}\)

def RK6_5(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = phi( tt[n]      , uu[n] )
        k2 = phi( tt[n]+h/4  , uu[n]+h*k1/4 )
        k3 = phi( tt[n]+h/4  , uu[n]+h*(k1+k2)/8 )
        k4 = phi( tt[n]+h/2  , uu[n]+h*(-k2+2*k3)/2 )
        k5 = phi( tt[n]+h*3/4, uu[n]+h*(3*k1+9*k4)/16 )
        k6 = phi( tt[n+1]    , uu[n]+h*(-3*k1+2*k2+12*k3-12*k4+8*k5)/7 )
        uu[n+1] = uu[n] + (7*k1+32*k3+12*k4+32*k5+7*k6)*h/90
    return uu  

2.12 Schéma de Runge-Kutta RK7_6 (Butcher à 7 étages d’ordre 6)

\( \begin{array}{c|ccccccc} 0 & 0 & 0 & 0 & 0 &0&0&0 \\ \frac{1}{2} & \frac{1}{2} & 0&0&0&0&0&0\\ \frac{2}{3} & \frac{2}{9} &\frac{4}{9}&0&0&0&0&0\\ \frac{1}{3} & \frac{7}{36} &\frac{2}{9}&-\frac{1}{12}&0&0&0&0\\ \frac{5}{6} & -\frac{35}{144} &-\frac{55}{36}&\frac{35}{48}&\frac{15}{8}&0&0&0\\ \frac{1}{6} & -\frac{1}{360} &-\frac{11}{36}&-\frac{1}{8}&\frac{1}{2}&\frac{1}{10}&0&0\\ 1 & \frac{-41}{260} &\frac{22}{13}&\frac{43}{156}&-\frac{118}{39}&\frac{32}{195}&\frac{80}{39}&0\\ \hline & \frac{13}{200} & 0&\frac{11}{40} & \frac{11}{40} & \frac{4}{25} & \frac{4}{25} & \frac{13}{200} \end{array}\)

def RK7_6(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = phi( tt[n]      , uu[n] )
        k2 = phi( tt[n]+h/2  , uu[n]+h*(        k1                                                     )/2 )
        k3 = phi( tt[n]+h*2/3, uu[n]+h*(      2*k1     +4*k2                                           )/9 )
        k4 = phi( tt[n]+h/3  , uu[n]+h*(      7*k1     +8*k2      -3*k3                                )/36 )
        k5 = phi( tt[n]+h*5/6, uu[n]+h*(    -35*k1   -220*k2    +105*k3    +270*k4                     )/144 )
        k6 = phi( tt[n]+h/6  , uu[n]+h*(       -k1   -110*k2     -45*k3    +180*k4     +36*k5          )/360 )
        k7 = phi( tt[n+1]    , uu[n]+h*(-41/260*k1 +22/13*k2 +43/156*k3 -118/39*k4 +32/195*k5 +80/39*k6)     )
        uu[n+1] = uu[n] + (13/200*k1+11/40*k3+11/40*k4+4/25*k5+4/25*k6+13/200*k7)*h
    return uu  

3 Schémas implicites

3.1 Schéma d’Euler régressif

\( \begin{cases} u_0=y_0,\\ u_{n+1}=u_n+h\varphi(t_{n+1},u_{n+1})& n=0,1,2,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) zéro de la fonction

\( x\mapsto -x+u_n+h\varphi(t_{n+1},x). \)

def EI(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        temp = fsolve(lambda x: -x+uu[n]+h*phi(tt[n+1],x), uu[n])
        uu[n+1] = temp[0]
    return uu

3.2 Schéma de Crank-Nicolson

\( \begin{cases} u_0=y_0,\\ u_{n+1}=u_n+\frac{h}{2}\Bigl( \varphi_n +\varphi(t_{n+1},u_{n+1})\Bigr)& n=0,1,2,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_n+\frac{h}{2}( \varphi_n +\varphi(t_{n+1},x)). \)

def CN(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        temp = fsolve(lambda x: -x+uu[n]+0.5*h*( phi(tt[n+1],x)+phi(tt[n],uu[n]) ), uu[n])
        uu[n+1] = temp[0]
    return uu

3.3 Schéma AM-2

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{n+1}=u_n+\frac{h}{12}\Bigl(5\varphi(t_{n+1},u_{n+1})+8 \varphi_n - \varphi_{n-1} \Bigr)& n=1,2,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_n+\frac{h}{12}\Bigl(5\varphi(t_{n+1},x)+8 \varphi_n - \varphi_{n-1} \Bigr). \)

def AM2(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 2
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        temp = fsolve(lambda x: -x+uu[n]+h*( 5*phi(tt[n+1],x)+8*phi(tt[n],uu[n])-phi(tt[n-1],uu[n-1]) )/12, uu[n])
        uu[n+1] = temp[0]
    return uu

3.4 Schéma AM-3

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{2}=y_2,\\ u_{n+1}=u_n+\frac{h}{24}\Bigl(9\varphi(t_{n+1},u_{n+1})+19 \varphi_n -5 \varphi_{n-1} + \varphi_{n-2} \Bigr)& n=2,3,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_n+\frac{h}{24}\Bigl(9\varphi(t_{n+1},x)+19 \varphi_n -5 \varphi_{n-1} + \varphi_{n-2} \Bigr). \)

def AM3(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 3
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        temp = fsolve(lambda x: -x+uu[n]+h*( 9*phi(tt[n+1],x)+19*phi(tt[n],uu[n])-5*phi(tt[n-1],uu[n-1])+phi(tt[n-2],uu[n-2]) )/24, uu[n])
        uu[n+1] = temp[0]
    return uu

3.5 Schéma AM-4

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{2}=y_2,\\ u_{3}=y_3,\\ u_{n+1}=u_n+\frac{h}{720}\Bigl(251\varphi(t_{n+1},u_{n+1})+646 \varphi_n -264 \varphi_{n-1} +106 \varphi_{n-2} -19 \varphi_{n-3} \Bigr)& n=3,4,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_n+\frac{h}{720}\Bigl(251\varphi(t_{n+1},x)+646 \varphi_n -264 \varphi_{n-1} +106 \varphi_{n-2} -19 \varphi_{n-3} \Bigr). \)

def AM4(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 4
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        temp = fsolve(lambda x: -x+uu[n]+h*( 251*phi(tt[n+1],x)+646*phi(tt[n],uu[n])-264*phi(tt[n-1],uu[n-1])+106*phi(tt[n-2],uu[n-2])-19*phi(tt[n-3],uu[n-3]) )/720, uu[n])
        uu[n+1] = temp[0]
    return uu

3.6 Schéma AM-5

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{2}=y_2,\\ u_{3}=y_3,\\ u_{4}=y_4,\\ u_{n+1}=u_n+\frac{h}{1440}\Bigl(475\varphi(t_{n+1},u_{+1})+1427 \varphi_n -798 \varphi_{n-1} +482 \varphi_{n-2} -173 \varphi_{n-3} +27 \varphi_{n-4} \Bigr),& n=4,5,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_n+\frac{h}{1440}\Bigl(475\varphi(t_{n+1},x)+1427 \varphi_n -798 \varphi_{n-1} +482 \varphi_{n-2} -173 \varphi_{n-3} +27 \varphi_{n-4} \Bigr). \)

def AM5(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 5
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        temp = fsolve(lambda x: -x+uu[n]+h*( 475*phi(tt[n+1],x)+1427*phi(tt[n],uu[n])-798*phi(tt[n-1],uu[n-1])+482*phi(tt[n-2],uu[n-2])-173*phi(tt[n-3],uu[n-3])+27*phi(tt[n-4],uu[n-4]))/1440, uu[n])
        uu[n+1] = temp[0]
    return uu

3.7 Schéma MS-2

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{n+1}=u_{n-1}+\frac{h}{3}\Bigl(\varphi(t_{n+1},u_{n+1})+4 \varphi_n + \varphi_{n-1} \Bigr)& n=1,2,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_{n-1}+\frac{h}{3}\Bigl(\varphi(t_{n+1},x)+4 \varphi_n + \varphi_{n-1} \Bigr). \)

def MS2(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 2
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        k0 = phi( tt[n], uu[n] )
        k1 = phi( tt[n-1], uu[n-1] )
        temp = fsolve(lambda x: -x+uu[n-1]+h*(phi(tt[n+1],x)+4*k0+k1) )/3, uu[n])
        uu[n+1] = temp[0]
    return uu

SyntaxError: unmatched ')' (1737807791.py, line 11)

3.8 Schéma BDF2

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{n+1}=\frac{4}{3}u_n-\frac{1}{3}u_{n-1}+\frac{2}{3}h\varphi(t_{n+1},u_{n+1})& n=1,2,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+\frac{4}{3}u_n-\frac{1}{3}u_{n-1}+\frac{2}{3}h\varphi(t_{n+1},x). \)

def BDF2(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 2
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        temp = fsolve(lambda x: -x+4/3*uu[n]-1/3*uu[n-1] + 2/3*h*phi(tt[n+1],x) , uu[n])
        uu[n+1] = temp[0]
    return uu   

3.9 Schéma BDF3

\( \begin{cases} u_0=y_0,\\ u_1=y_1,\\ u_{2}=y_2,\\ u_{n+1}=\frac{18}{11}u_n-\frac{9}{11}u_{n-1}+\frac{2}{11}u_{n-2}+\frac{6}{11}h\varphi(t_{n+1},u_{n+1})& n=2,3,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+\frac{18}{11}u_n-\frac{9}{11}u_{n-1}+\frac{2}{11}u_{n-2}+\frac{6}{11}h\varphi(t_{n+1},x). \)

def BDF3(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    q = 3
    uu[:q] = sol_exacte(tt[:q])
    # récurrence
    for n in range(q-1,len(tt)-1):
        temp = fsolve(lambda x: -x+18/11*uu[n]-9/11*uu[n-1] + 2/11*uu[n-2]+6/11*h*phi(tt[n+1],x) , uu[n])
        uu[n+1] = temp[0]
    return uu

3.10 Schéma RK1_M

\( \begin{cases} u_0=y_0,\\ u_{n+1}=u_n+h\varphi\left(t_n+\frac{h}{2},\frac{u_n+u_{n+1}}{2}\right)& n=0,1,2,\dots N-1 \end{cases} \)

avec \(u_{n+1}\) un zéro de la fonction

\( x\mapsto -x+u_n+h\varphi\left(t_n+\frac{h}{2},\frac{u_n+x}{2}\right). \)

Rappelons-nous que ce schéma est bien un schéma de Runge-Kutta:

\( \begin{array}{c|cc} \frac{1}{2} & \frac{1}{2}\\ \hline & 1 \end{array} \quad \implies \begin{cases} u_0 = y_0 \\ K_1 = \varphi\left(t_n+\frac{h}{2},u_n+\frac{h}{2}K_1\right)\\ u_{n+1} = u_n + h K_1 & n=0,1,\dots N-1 \end{cases} \)

avec \(K_1\) un zéro de la fonction

\( x\mapsto -x+\varphi\left(t_n+\frac{h}{2},u_n+\frac{h}{2}x\right). \)

# Version "classique"
# def RK1_M(phi,tt,sol_exacte):
#     h  = tt[1]-tt[0]
#     uu = np.zeros_like(tt)
#     # initialisation
#     uu[0] = sol_exacte(tt[0])
#     # récurrence
#     for n in range(len(tt)-1):
#         uu[n+1] = fsolve(lambda x: -x+uu[n]+h*phi( (tt[n]+tt[n+1])/2,(uu[n]+x)/2 ), uu[n])[0]
#     return uu

# Version "Runge-Kutta"
def RK1_M(phi,tt,sol_exacte):
    h  = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = fsolve( lambda x : -x + phi(tt[n]+h/2, uu[n]+h*x/2), phi(tt[n],uu[n]) )[0]    
        uu[n+1] = uu[n]+h*k1
    return uu

3.11 Schéma Randau-IIa-2

\( \begin{array}{c|cc} \frac{1}{3} & \frac{5}{12} & -\frac{1}{12}\\ 1 & \frac{3}{4} & \frac{1}{4}\\ \hline & \frac{3}{4} & \frac{1}{4} \end{array} \quad \implies \begin{cases} u_0 = y_0 \\ K_1 = \varphi\left(t_{n}+\frac{1}{3}h,u_n+h\big(\frac{5}{12}K_1-\frac{1}{12}K_2\big)\right)\\ K_2 = \varphi\left(t_{n+1},u_n+h\big(\frac{3}{4}K_1+\frac{1}{4}K_2\big)\right)\\ u_{n+1} = u_n + \frac{h}{4}(3K_1+K_2) & n=0,1,\dots N-1 \end{cases} \)

avec \([K_1,K_2]\) un zéro de la fonction vectorielle

\( (x_1,x_2)\mapsto \begin{pmatrix} -x_1+\varphi\left(t_n+\frac{h}{3},u_n+h\big(\frac{5}{12}x_1-\frac{1}{12}x_2\big)\right)\\ -x_2+\varphi\left(t_n+\frac{h}{3},u_n+h\big(\frac{3}{4}x_1-\frac{1}{4}x_2\big)\right) \end{pmatrix} \).

def Randau_IIa_2(phi,tt,sol_exacte):
    h  = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        systeme = lambda KK : [ -KK[0] + phi( tt[n]+h/3, uu[n]+h/12*(5*KK[0]-KK[1]) ), 
                                -KK[1] + phi( tt[n+1]  , uu[n]+ h/4*(3*KK[0]+KK[1]) ) ]
        k_init = phi(tt[n],uu[n])
        k1,k2 = fsolve( systeme, (k_init,k_init) )    
        uu[n+1] = uu[n]+h/4*(3*k1+k2)
    return uu

4 Schémas predicteur-correcteur

4.1 Schéma de Heun

\( \begin{cases} u_0=y_0,\\ \tilde u = u_n+h \varphi_n \\ u_{n+1}=u_n+\frac{h}{2}\Bigl( \varphi_n +\varphi(t_{n+1},\tilde u)\Bigr)& n=0,1,2,\dots N-1 \end{cases} \)

Le schéma de Heun est à la fois un schéma prédicteur-correcteur et un schéma de Runge-Kutta.

Comme predictor-corrector, il utilise le schéma d’Euler explicite pour prédire la valeur de \(u_{n+1}\) et le schéma de Crank-Nicolson pour corriger cette valeur.

Comme schéma de Runge-Kutta, il s’écrit comme \( \begin{cases} u_0 = y_0,\\ k_1 = \varphi_n ,\\ k_2 = \varphi(t_{n+1},u_n+hk_1)\\ u_{n+1}=u_n+\frac{h}{2}\Bigl(k_1+k_2\Bigr)& n=0,1,2,\dots N-1 \end{cases} \) dont la matrice de Buthcher est \( \begin{array}{c|cc} 0 & 0 & 0\\ 1 & 1 & 0\\ \hline & \frac{1}{2} & \frac{1}{2} \end{array} \)

def heun(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[0] = sol_exacte(tt[0])
    # récurrence
    for n in range(len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n+1], uu[n] + k1*h  )
        uu[n+1] = uu[n] + (k1+k2)*h/2
    return uu

4.2 Schéma AM-4 AB-2/3/4/5

• Predictor : Adam-Bashforth à 2, 3, 4 et 5 pas respectivement
• Corrector : Adam-Moulton à 4 pas

def AM4AB2(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[:4] = sol_exacte(tt[:4])
    # récurrence
    for n in range(3,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        pred = uu[n] + (3*k1-k2)*h/2
        uu[n+1] = uu[n]+h*( 251*phi(tt[n+1],pred)+646*phi(tt[n],uu[n])-264*phi(tt[n-1],uu[n-1])+106*phi(tt[n-2],uu[n-2])-19*phi(tt[n-3],uu[n-3]) )/720
    return uu

def AM4AB3(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[:4] = sol_exacte(tt[:4])
    # récurrence
    for n in range(3,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        pred = uu[n] + (23*k1-16*k2+5*k3)*h/12
        uu[n+1] = uu[n]+h*( 251*phi(tt[n+1],pred)+646*phi(tt[n],uu[n])-264*phi(tt[n-1],uu[n-1])+106*phi(tt[n-2],uu[n-2])-19*phi(tt[n-3],uu[n-3]) )/720
    return uu

def AM4AB4(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[:4] = sol_exacte(tt[:4])
    # récurrence
    for n in range(3,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        k4 = phi( tt[n-3], uu[n-3] )
        pred = uu[n] + (55*k1-59*k2+37*k3-9*k4)*h/24
        uu[n+1] = uu[n]+h*( 251*phi(tt[n+1],pred)+646*phi(tt[n],uu[n])-264*phi(tt[n-1],uu[n-1])+106*phi(tt[n-2],uu[n-2])-19*phi(tt[n-3],uu[n-3]) )/720
    return uu

def AM4AB5(phi,tt,sol_exacte):
    h = tt[1]-tt[0]
    uu = np.zeros_like(tt)
    # initialisation
    uu[:5] = sol_exacte(tt[:5])
    # récurrence
    for n in range(4,len(tt)-1):
        k1 = phi( tt[n], uu[n] )
        k2 = phi( tt[n-1], uu[n-1] )
        k3 = phi( tt[n-2], uu[n-2] )
        k4 = phi( tt[n-3], uu[n-3] )
        k5 = phi( tt[n-4], uu[n-4] )
        pred = uu[n] + (1901*k1-2774*k2+2616*k3-1274*k4+251*k5)*h/720
        uu[n+1] = uu[n]+h*( 251*phi(tt[n+1],pred)+646*phi(tt[n],uu[n])-264*phi(tt[n-1],uu[n-1])+106*phi(tt[n-2],uu[n-2])-19*phi(tt[n-3],uu[n-3]) )/720
    return uu

5 Convergence

On initialise le problème de Cauchy, on définit l’équation différentielle (phi est une fonction python qui contient la fonction mathématique \(\varphi(t, y)\) dépendant des variables \(t\) et \(y\)) et enfin on définit la solution exacte:

t0, y0, tfinal = 0, 1, 3

# CHOIX DU CAS TEST

sol_exacte = lambda t: np.exp(t)
phi = lambda t,y: y

# Autres cas tests
# ==========================================
# sol_exacte = lambda t: np.exp(-t)
# phi = lambda t,y: -y

# sol_exacte = lambda t: np.sqrt(2*t+1)
# phi = lambda t,y: 1/y

# sol_exacte = lambda t: np.sqrt(t**2+1)
# phi = lambda t,y: t/y

# sol_exacte = lambda t: 1/np.sqrt(1-2*t)
# phi = lambda t,y: y**3

Pour chaque schéma, on calcule la solution approchée avec différentes valeurs de \(h_k=1/N_k\); on sauvegarde les valeurs de \(h_k\) dans le vecteur H.

Pour chaque valeur de \(h_k\), on calcule le maximum de la valeur absolue de l’erreur et on sauvegarde toutes ces erreurs dans le vecteur err_schema de sort que err_schema[k] contient \(e_k=\max_{i=0,...,N_k}|y(t_i)-u_{i}|\).

Nous pouvons utiliser deux méthodes différentes pour stocker ces informations.

5.1 Méthode 1

La première méthode est celle utilisée lors des deux premiers TP: on crée autant de listes que de solutions approchées.

H = []

err_ep   = []
err_AB2  = []
err_AB3  = []
err_AB4  = []
err_AB5  = []
err_N2   = []
err_N3   = []
err_N4   = []
err_RK4  = []
err_RK6_5  = []
err_RK7_6  = []

err_er   = []
err_CN   = []
err_AM2  = []
err_AM3  = []
err_AM4  = []
err_AM5  = []
err_BDF2 = []
err_BDF3 = []
err_MS2=[]
err_RK1_M = []
err_Randau_IIa_2 = []

err_em   = []
err_heun = []
err_AM4AB2  = []
err_AM4AB3  = []
err_AM4AB4  = []
err_AM4AB5  = []

N = 10
for k in range(6):
    N += 50 #2**(k+3)
    tt = np.linspace(t0,tfinal,N+1)
    h = tt[1]-tt[0]
    H.append(h)
    yy = np.array([sol_exacte(t) for t in tt])
    
    # schemas explicites
    uu_ep   = EE(phi,tt,sol_exacte)
    uu_AB2  = AB2(phi,tt,sol_exacte)
    uu_AB3  = AB3(phi,tt,sol_exacte)
    uu_AB4  = AB4(phi,tt,sol_exacte)
    uu_AB5  = AB5(phi,tt,sol_exacte)
    uu_N2   = N2(phi,tt,sol_exacte)
    uu_N3   = N3(phi,tt,sol_exacte)
    uu_N4   = N4(phi,tt,sol_exacte)
    uu_em   = EM(phi,tt,sol_exacte)
    uu_RK4  = RK4(phi,tt,sol_exacte)
    uu_RK6_5  = RK6_5(phi,tt,sol_exacte)
    uu_RK7_6  = RK7_6(phi,tt,sol_exacte)
    # schemas implicites
    uu_er   = EI(phi,tt,sol_exacte)
    uu_CN   = CN(phi,tt,sol_exacte)
    uu_AM2  = AM2(phi,tt,sol_exacte)
    uu_AM3  = AM3(phi,tt,sol_exacte)
    uu_AM4  = AM4(phi,tt,sol_exacte)
    uu_AM5  = AM5(phi,tt,sol_exacte)
    uu_BDF2 = BDF2(phi,tt,sol_exacte)
    uu_BDF3 = BDF3(phi,tt,sol_exacte)
    uu_MS2  = MS2(phi,tt,sol_exacte)
    uu_RK1_M = RK1_M(phi,tt,sol_exacte)
    uu_Randau_IIa_2 = Randau_IIa_2(phi,tt,sol_exacte)
    # schemas predictor-corrector
    uu_heun   = heun(phi,tt,sol_exacte)
    uu_AM4AB2 = AM4AB2(phi,tt,sol_exacte)
    uu_AM4AB3 = AM4AB3(phi,tt,sol_exacte)
    uu_AM4AB4 = AM4AB4(phi,tt,sol_exacte)
    uu_AM4AB5 = AM4AB5(phi,tt,sol_exacte)

    # erreurs
    err_ep.append(np.linalg.norm(uu_ep-yy,np.inf))
    err_AB2.append(np.linalg.norm(uu_AB2-yy,np.inf))
    err_AB3.append(np.linalg.norm(uu_AB3-yy,np.inf))
    err_AB4.append(np.linalg.norm(uu_AB4-yy,np.inf))
    err_AB5.append(np.linalg.norm(uu_AB5-yy,np.inf))
    err_N2.append(np.linalg.norm(uu_N2-yy,np.inf))
    err_N3.append(np.linalg.norm(uu_N3-yy,np.inf))
    err_N4.append(np.linalg.norm(uu_N4-yy,np.inf))
    err_em.append(np.linalg.norm(uu_em-yy,np.inf))
    err_RK4.append(np.linalg.norm(uu_RK4-yy,np.inf))
    err_RK6_5.append(np.linalg.norm(uu_RK6_5-yy,np.inf))
    err_RK7_6.append(np.linalg.norm(uu_RK7_6-yy,np.inf))
    
    err_er.append(np.linalg.norm(uu_er-yy,np.inf))
    err_CN.append(np.linalg.norm(uu_CN-yy,np.inf))
    err_AM2.append(np.linalg.norm(uu_AM2-yy,np.inf))
    err_AM3.append(np.linalg.norm(uu_AM3-yy,np.inf))
    err_AM4.append(np.linalg.norm(uu_AM4-yy,np.inf))
    err_AM5.append(np.linalg.norm(uu_AM5-yy,np.inf))
    err_BDF2.append(np.linalg.norm(uu_BDF2-yy,np.inf))
    err_BDF3.append(np.linalg.norm(uu_BDF3-yy,np.inf))
    err_MS2.append(np.linalg.norm(uu_MS2-yy,np.inf))
    err_RK1_M.append(np.linalg.norm(uu_RK1_M-yy,np.inf))
    err_Randau_IIa_2.append(np.linalg.norm(uu_Randau_IIa_2-yy,np.inf))
    
    err_heun.append(np.linalg.norm(uu_heun-yy,np.inf))
    err_AM4AB2.append(np.linalg.norm(uu_AM4AB2-yy,np.inf))
    err_AM4AB3.append(np.linalg.norm(uu_AM4AB3-yy,np.inf))
    err_AM4AB4.append(np.linalg.norm(uu_AM4AB4-yy,np.inf))
    err_AM4AB5.append(np.linalg.norm(uu_AM4AB5-yy,np.inf))

Pour estimer l’ordre de convergence on estime la pente de la droite qui relie l’erreur au pas \(k\) à l’erreur au pas \(k+1\) en echelle logarithmique en utilisant la fonction polyfit basée sur la régression linéaire.

# print (f'EE    {np.polyfit(np.log(H),np.log(err_ep),    1)[0]:1.2f}' )
# print (f'AB2   {np.polyfit(np.log(H),np.log(err_AB2),   1)[0]:1.2f}'  )
# print (f'AB3   {np.polyfit(np.log(H),np.log(err_AB3),   1)[0]:1.2f}' )
# print (f'AB4   {np.polyfit(np.log(H),np.log(err_AB4),   1)[0]:1.2f}' )
# print (f'AB5   {np.polyfit(np.log(H),np.log(err_AB5),   1)[0]:1.2f}' )
# print (f'N2    {np.polyfit(np.log(H),np.log(err_N2),    1)[0]:1.2f}' )
# print (f'N3    {np.polyfit(np.log(H),np.log(err_N3),    1)[0]:1.2f}' )
# print (f'N4    {np.polyfit(np.log(H),np.log(err_N4),    1)[0]:1.2f}' )
# print (f'EM    {np.polyfit(np.log(H),np.log(err_em),    1)[0]:1.2f}' )
# print (f'RK4   {np.polyfit(np.log(H),np.log(err_RK4),   1)[0]:1.2f}' )
# print (f'RK6_5 {np.polyfit(np.log(H),np.log(err_RK6_5), 1)[0]:1.2f}' )
# print (f'RK7_6 {np.polyfit(np.log(H),np.log(err_RK7_6), 1)[0]:1.2f}' )
# print('\n')
# print (f'EI   {np.polyfit(np.log(H),np.log(err_er),   1)[0]:1.2f}' )
# print (f'CN   {np.polyfit(np.log(H),np.log(err_CN),   1)[0]:1.2f}' )
# print (f'AM2  {np.polyfit(np.log(H),np.log(err_AM2),  1)[0]:1.2f}' )
# print (f'AM3  {np.polyfit(np.log(H),np.log(err_AM3),  1)[0]:1.2f}' )
# print (f'AM4  {np.polyfit(np.log(H),np.log(err_AM4),  1)[0]:1.2f}' )
# print (f'AM5  {np.polyfit(np.log(H),np.log(err_AM5),  1)[0]:1.2f}' )
# print (f'BDF2 {np.polyfit(np.log(H),np.log(err_BDF2), 1)[0]:1.2f}' ) 
# print (f'BDF3 {np.polyfit(np.log(H),np.log(err_BDF3), 1)[0]:1.2f}' )
# print (f'MS2  {np.polyfit(np.log(H),np.log(err_MS2),  1)[0]:1.2f}' )
# print (f'RK1_M {np.polyfit(np.log(H),np.log(err_RK1_M),  1)[0]:1.2f}' )
# print (f'Randau_IIa_2 {np.polyfit(np.log(H),np.log(err_Randau_IIa_2),  1)[0]:1.2f}' )
# print('\n')
# print (f'Heun    {np.polyfit(np.log(H),np.log(err_heun),   1)[0]:1.2f}' )
# print (f'AM4_AB2 {np.polyfit(np.log(H),np.log(err_AM4AB2), 1)[0]:1.2f}' )
# print (f'AM4_AB3 {np.polyfit(np.log(H),np.log(err_AM4AB3), 1)[0]:1.2f}' )
# print (f'AM4_AB4 {np.polyfit(np.log(H),np.log(err_AM4AB4), 1)[0]:1.2f}' )
# print (f'AM4_AB5 {np.polyfit(np.log(H),np.log(err_AM4AB5), 1)[0]:1.2f}' )

Pour afficher l’ordre de convergence on utilise une échelle logarithmique : on représente \(\ln(h)\) sur l’axe des abscisses et \(\ln(\text{err})\) sur l’axe des ordonnées. Le but de cette représentation est clair: si \(\text{err}=Ch^p\) alors \(\ln(\text{err})=\ln(C)+p\ln(h)\). En échelle logarithmique, \(p\) représente donc la pente de la ligne droite \(\ln(\text{err})\).

plt.figure(figsize=(24,7))

plt.subplot(1,3,1)
plt.loglog(H,err_ep,    'r-o',label=f'AB-1 = EE  {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_ep),    <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AB2,   'g-+',label=f'AB2   {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AB2),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AB3,   'c-D',label=f'AB3   {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AB3),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AB4,   'y-*',label=f'AB4   {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AB4),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AB5,   'r-.',label=f'AB5   {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AB5),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_N2,    'y-*',label=f'N2    {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_N2),    <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_N3,    'r-<',label=f'N3    {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_N3),    <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_N4,    'b-+',label=f'N4    {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_N4),    <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_em,    'c-o',label=f'EM    {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_em),    <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_RK4,   'g-o',label=f'RK4_1 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_RK4),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_RK6_5, 'b-*',label=f'RK6_5 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_RK6_5), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_RK7_6, 'r-D',label=f'RK7_6 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_RK7_6), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas explicites")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

plt.subplot(1,3,2)
plt.loglog(H,err_er,  'r-o',label=f'AM-0 = EI   {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_er),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_CN,  'b-v',label=f'AM-1 = CN   {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_CN),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM2, 'm->',label=f'AM2  {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM2),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM3, 'c-D',label=f'AM3  {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM3),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM4, 'g-+',label=f'AM4  {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM4),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM5, 'b->',label=f'AM5  {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM5),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_BDF2,'y-*',label=f'BDF2 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_BDF2), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_BDF3,'r-<',label=f'BDF3 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_BDF3), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_MS2, 'm-o',label=f'MS2  {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_MS2),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_RK1_M, 'c-+',label=f'RK1_M {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_RK1_M),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_Randau_IIa_2, 'g-D',label=f'Randau_IIa_2 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_Randau_IIa_2),  <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas implicites")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

plt.subplot(1,3,3)
plt.loglog(H,err_heun,   'y->',label=f'Heun    {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_heun),   <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM4AB2, 'r-<',label=f'AM4_AB2 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM4AB2), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM4AB3, 'b-v',label=f'AM4_AB3 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM4AB3), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}' )
plt.loglog(H,err_AM4AB4, 'm-*',label=f'AM4_AB4 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM4AB4), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}')
plt.loglog(H,err_AM4AB5, 'g-+',label=f'AM4_AB5 {</span>np<span class="sc">.</span>polyfit(np.log(H),np.log(err_AM4AB5), <span class="dv">1</span>)[<span class="dv">0</span>]<span class="sc">:1.2f}' )
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas predicteur-correcteur")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

Si on note

• \(\omega_{[C]}\) l’ordre du corrector (schéma implicite)
• \(\omega_{[P]}\) l’ordre du predictor (schéma explicite)

alors l’ordre du schéma predictor-corrector est \(\omega_{[PC]}=\min\{\omega_{[C]},\omega_{[P]}+1\}\)

Pour les schémas AM4-ABx, le schéma corrector est d’ordre \(p=5\), pour ne pas perdre en ordre convergence il faut choisir un schéma predictor d’ordre \(p-1\) (ici AB4).

5.2 Méthode 2 : version compacte

De manière compacte en utilisant une liste des noms des schémas et des dictionnaires:

schemasEXPL = ['EE','AB2','AB3','AB4','AB5','N2','N3','N4', 'EM', 'RK4', 'RK6_5', 'RK7_6']
schemasIMPL = ['EI', 'CN', 'AM2', 'AM3', 'AM4', 'AM5', 'BDF2', 'BDF3', 'MS2', 'RK1_M', 'Randau_IIa_2']
schemasPRCO = ['heun', 'AM4AB2', 'AM4AB3', 'AM4AB4', 'AM4AB5']

schemas = schemasEXPL + schemasIMPL + schemasPRCO

H = []
err = { schemas[s] : [] <span class="cf">for</span> s <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(schemas)) }

N = 10
for k in range(6):
    N += 50
    tt = np.linspace(t0, tfinal, N+1)
    h = tt[1]-tt[0]
    H.append(h)
    yy = np.array([sol_exacte(t) for t in tt])
    uu = { s : <span class="bu">eval</span>(s)(phi, tt, sol_exacte) <span class="cf">for</span> s <span class="kw">in</span> schemas }
    for key in uu:
        err[key].append(np.linalg.norm(uu[key]-yy,np.inf))

ordre = {  key : np.polyfit(np.log(H),np.log(err[key]),<span class="dv">1</span>)[<span class="dv">0</span>] <span class="cf">for</span> key <span class="kw">in</span> err  }

    
plt.figure(figsize=(24,7))
markers=['^', 's', 'p', 'h', '8', 'D', '>', '<', '*','+','o', 'x']

plt.subplot(1,3,1)
for idx,s in enumerate(schemasEXPL):
    plt.loglog(H,err[s], label=f'{</span>s<span class="sc">}: {</span>ordre[s]<span class="sc">:1.2f}',marker=markers[idx])
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas explicites")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

plt.subplot(1,3,2)
for idx,s in enumerate(schemasIMPL):
    plt.loglog(H,err[s], label=f'{</span>s<span class="sc">}: {</span>ordre[s]<span class="sc">:1.2f}',marker=markers[idx])
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas implicites")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

plt.subplot(1,3,3)
for idx,s in enumerate(schemasPRCO):
    plt.loglog(H,err[s], label=f'{</span>s<span class="sc">}: {</span>ordre[s]<span class="sc">:1.2f}',marker=markers[idx])
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas predicteur-correcteur")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, ncol=1)
plt.grid(True);

5.3 Méthode 2 bis : version compacte

Idem sans utiliser eval :

schemasEXPL = [EE, AB2, AB3, AB4, AB5, N2, N3, N4, EM, RK4, RK6_5, RK7_6]
schemasIMPL = [EI, CN, AM2, AM3, AM4, AM5, BDF2, BDF3, MS2, RK1_M, Randau_IIa_2]
schemasPRCO = [heun, AM4AB2, AM4AB3, AM4AB4, AM4AB5]

schemas = schemasEXPL + schemasIMPL + schemasPRCO

H = []
err = { sfct.<span class="va">__name__</span> : [] <span class="cf">for</span> sfct <span class="kw">in</span> schemas }

N = 10
for k in range(6):
    N += 50
    tt = np.linspace(t0, tfinal, N+1)
    h = tt[1]-tt[0]
    H.append(h)
    yy = np.array([sol_exacte(t) for t in tt])
    uu = { sfct.<span class="va">__name__</span> : sfct(phi, tt, sol_exacte) <span class="cf">for</span> sfct <span class="kw">in</span> schemas }
    for key in uu:
        err[key].append(np.linalg.norm(uu[key]-yy,np.inf))

ordre = {  key : np.polyfit(np.log(H),np.log(err[key]),<span class="dv">1</span>)[<span class="dv">0</span>] <span class="cf">for</span> key <span class="kw">in</span> err  }

    
plt.figure(figsize=(24,7))
markers=['^', 's', 'p', 'h', '8', 'D', '>', '<', '*','+','o', 'x']

plt.subplot(1,3,1)
for idx,sfct in enumerate(schemasEXPL):
    s = sfct.__name__
    plt.loglog(H,err[s], label=f'{</span>s<span class="sc">}: {</span>ordre[s]<span class="sc">:1.2f}',marker=markers[idx])
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas explicites")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

plt.subplot(1,3,2)
for idx,sfct in enumerate(schemasIMPL):
    s = sfct.__name__
    plt.loglog(H,err[s], label=f'{</span>s<span class="sc">}: {</span>ordre[s]<span class="sc">:1.2f}',marker=markers[idx])
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas implicites")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, shadow=True, ncol=1)
plt.grid(True)

plt.subplot(1,3,3)
for idx,sfct in enumerate(schemasPRCO):
    s = sfct.__name__
    plt.loglog(H,err[s], label=f'{</span>s<span class="sc">}: {</span>ordre[s]<span class="sc">:1.2f}',marker=markers[idx])
plt.xlabel('$h$')
plt.ylabel('$e$')
plt.title("Schemas predicteur-correcteur")
plt.legend(loc='upper center', bbox_to_anchor=(0.5, -0.05), fancybox=True, ncol=1)
plt.grid(True);