From a7db4a4a59e1759af59e0d82092b83eebd007856 Mon Sep 17 00:00:00 2001
From: Eric Dalissier <Eric.Dalissier@imag.fr>
Date: Fri, 30 Nov 2012 16:18:21 +0000
Subject: [PATCH] Rk3 add, and documentation on stretching

---
 Docs/Stretching.pdf                           | Bin 0 -> 740328 bytes
 Docs/Stretching.tex                           | 353 ++++++++++++++++++
 Docs/images/Euler.png                         | Bin 0 -> 5667 bytes
 Docs/images/RK2.png                           | Bin 0 -> 5956 bytes
 Docs/images/RK2_error.png                     | Bin 0 -> 131071 bytes
 Docs/images/RK3.png                           | Bin 0 -> 6733 bytes
 Docs/images/RK3_TVD_error.png                 | Bin 0 -> 133031 bytes
 Docs/images/RK3_error.png                     | Bin 0 -> 137438 bytes
 Docs/images/RK4.png                           | Bin 0 -> 6427 bytes
 Docs/images/RK4_error.png                     | Bin 0 -> 145770 bytes
 HySoP/hysop/operator/differentialOperator.py  |   5 +-
 .../hysop/operator/differentialOperator_d.py  |  95 ++++-
 HySoP/hysop/operator/stretching_d.py          |  75 +++-
 HySoP/hysop/particular_solvers/basic.py       |   2 +
 .../integrator/runge_kutta2stretching.py      |   3 +-
 .../integrator/runge_kutta3stretching.py      |  84 +++++
 .../integrator/runge_kutta4stretching.py      |   7 +-
 .../test/test_operator/test_DivProduct.py     |   8 +-
 .../test/test_operator/test_Stretching.py     |  44 ++-
 .../test/test_particular_solvers/test_RK.py   |  53 ++-
 20 files changed, 671 insertions(+), 58 deletions(-)
 create mode 100644 Docs/Stretching.pdf
 create mode 100644 Docs/Stretching.tex
 create mode 100644 Docs/images/Euler.png
 create mode 100644 Docs/images/RK2.png
 create mode 100644 Docs/images/RK2_error.png
 create mode 100644 Docs/images/RK3.png
 create mode 100644 Docs/images/RK3_TVD_error.png
 create mode 100644 Docs/images/RK3_error.png
 create mode 100644 Docs/images/RK4.png
 create mode 100644 Docs/images/RK4_error.png
 create mode 100644 HySoP/hysop/particular_solvers/integrator/runge_kutta3stretching.py

diff --git a/Docs/Stretching.pdf b/Docs/Stretching.pdf
new file mode 100644
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diff --git a/Docs/Stretching.tex b/Docs/Stretching.tex
new file mode 100644
index 000000000..6e3226adc
--- /dev/null
+++ b/Docs/Stretching.tex
@@ -0,0 +1,353 @@
+\documentclass[a4paper,10pt]{article}
+\usepackage[utf8x]{inputenc}
+\usepackage{amsmath}
+% \usepackage{showkeys} % affiche les labels et references
+\usepackage[T1]{fontenc}
+\usepackage{multirow}
+\usepackage{amsthm,amssymb}
+\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
+\usepackage{graphicx}
+\graphicspath{{./images/}}
+\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
+\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
+
+\usepackage[pdftex]{hyperref}
+\hypersetup{
+  pdftitle={Stretching-Stability},
+  pdfauthor={Dalissier Eric, Mimeau Chloe},
+  pdfcreator={PARMES},
+  pdfkeywords={Stretching, Stability},
+  pdfsubject={Vorticity}
+% pagebackref=false,
+%colorlinks=false,citecolor=red,
+%linkcolor=blue,
+%plainpages=false,
+%breaklinks=true,
+% pdfpagelabels=true
+}
+
+
+\theoremstyle{plain}
+\newtheorem{Theoreme}{Th\'eor\`eme}[section]
+%opening
+\title{Condition de Stabilit\'e sur le Stretching}
+\author{}
+\date{}
+
+\begin{document}
+
+\maketitle
+
+L'\'equation de Navier-Stokes peut \^etre r\'esolue en utilisant le splitting d'op\'e\-rateur, qui consiste \`a traiter ind\'ependamment les op\'erateurs du probl\`eme. Dans ce document, nous nous int\'eressons au terme de stretching qui repr\'esente les d\'eplac\'epements transversaux de l'\'ecoulement. L'\'equation de Stretching peut se pr\'esenter sous plusieurs formes~:
+\begin{align}
+\label{eq:gradstretch}
+ \frac{\partial \omega}{\partial t} = [\nabla u]. \omega \\
+\label{eq:transpgradstretch}
+ \frac{\partial \omega}{\partial t} =  [\nabla u]^T. \omega\\
+\label{eq:divstretch}
+ \frac{\partial \omega}{\partial t} = div( \omega u ) 
+\end{align}
+Une autre condition qui doit \^etre v\'erifi\'ee est la conservation de la masse, exprim\'ee par $div (u) =0$.
+
+La définition de la vorticit\'e est~:
+\begin{equation}
+ \omega = \overrightarrow{rot}(u)
+\end{equation}
+
+
+\section{Stabilit\'e}
+L'\'equation \eqref{eq:gradstretch} conserve la masse. Cette expression peut devenir instable num\'e\-riquement, tout d\'ependra du sch\'ema d'int\'egration en temps ainsi que de la forme de la matrice $\nabla u$.
+On peut se ramener \`a l'\'etude d'un syst\`eme lin\'eaire de la forme~:
+\begin{equation}
+\label{eq:problemeglobal}
+ \omega'=A \omega
+\end{equation}
+Dans le cas o\`u $A$ est diagonalisable, on peut se ramener \`a l'\'etude des valeurs propres de A, car l'on aura~:
+\begin{equation}
+\label{eq:eigentime}
+\omega' = \lambda_i \omega \hspace{0.2\linewidth} \text{ o\`u } i\in \{0,1,2\} 
+\end{equation}
+
+
+On va s'int\'eresser \`a l'\'etude de la condition \`a mettre sur $\lambda$. On veut pour que le syst\`eme converge que Re$(\lambda)<1$, car c'est elle qui va entra\^iner la convergence dans la r\'esolution de l'EDO. (il faut que la puissance de l'exponentielle soit plus petite que 1 pour converger) On ne s'int\'eresse qu'\`a la partie r\'eelle car la partie complexe est une fonction oscillante entre -1 et 1, et donc bornable.
+
+On veut que l'\'equation \eqref{eq:eigentime} converge en temps, c'est-\`a-dire pour le cas de sch\'ema \`a 2 pas de temps~:
+\begin{equation}
+ \omega^{n+1} = B \omega^n
+\end{equation}
+Que $B$ doit \^etre born\'ee en valeur absolue par 1 de tel sorte que~:
+\begin{equation}
+|\frac{\omega^{n+1}}{\omega^n}|<1
+\end{equation}
+
+Les conditions pour les diff\'erents sch\'emas sont les suivantes~:
+\begin{itemize}
+ \item Le sch\'ema d'Euler (ordre 1)\\
+L'\'equation \eqref{eq:problemeglobal} discr\'etis\'ee avec un sch\'ema d'Euler s'\'ecrit sous la forme~:
+\begin{equation}
+ \frac{\omega^{n+1}-\omega^{n}}{\Delta t} = \nabla u \omega^n
+\end{equation}
+cela nous donne le r\'esultat suivant~:
+\begin{equation}
+ \omega^{n+1}=(1+\nabla u. \Delta t) \omega^n
+\end{equation}
+La condition pour le sch\'ema d'Euler est alors que $|1+X|<1$, avec $X=\nabla u.\Delta t$.
+\begin{figure}[h]
+\centering
+ \includegraphics[width=0.6\linewidth]{Euler}
+\caption{Repr\'esentation graphique des \'equations $y=|1+x|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta  t$}
+\end{figure}
+Il faut donc que $X= \nabla u.\Delta t$ soit inf\'erieure \`a $2$.\\
+On aura donc :
+\begin{center}
+ $\nabla u. \Delta t <2$ 
+\end{center}
+En faisant le changement de variable, si la matrice $A=[\nabla u]$ est diagonalisable. Nous avons~:
+\begin{center}
+ $\lambda_i . \Delta t <2$\hspace{0.2\linewidth} pour $i=0,1,2$\\
+ $\Delta t < \frac{2}{\lambda_i}$
+\end{center}
+On utilise ensuite le th\'eor\`eme de Gerschgorin, qui permet de majorer les valeurs propres de la matrice $A$. On \'economise ainsi le nombre d'op\'eration \`a faire afin de d\'eterminer une condition pour le pas de temps $\Delta t$.
+\begin{Theoreme}[th\'eor\`eme de Gerschgorin\label{thm:gg}]
+Soit $A$ une matrice complexe de taille $n\times n$, de terme g\'en\'eral $a_{ij}$. Toute valeur propre de $A$ appartient au moins \`a un disque de Gerschgorin $\displaystyle D_i= \{z\in \mathbb{C}, |a_{ii}-z|\leq \sum_{i\neq j} |a_{ij}| \}$.
+\end{Theoreme}
+On peut donc majorer le rayon spectral de la matrice $A$ par $\displaystyle\max_i(\sum_{j} |a_{ij}|)$. On obtient ainsi la condition de stabilit\'e suivante pour le sch\'ema d'euler~:
+\begin{equation}
+ \Delta t < \dfrac{2}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
+\end{equation}
+
+ \item Le sch\'ema de Runge Kutta (ordre 2)
+Pour Runge-Kutta d'ordre 2, on obtient une majoration identique \`a celle d'Euler, car on a~:
+\begin{equation}
+ \omega^{n+1} = \bigl(1 + \nabla u.\Delta t + \frac{1}{2}(\nabla u.\Delta t)^2 \bigr)\ \omega^{n}
+\end{equation}
+\begin{figure}[ht]
+\centering
+ \includegraphics[width=0.6\linewidth]{RK2}
+\caption{Repr\'esentation graphique des \'equations $y=|1+x + \frac{1}{2}x^2|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta  t$}
+\end{figure}
+
+On obtient alors en appliquant aussi le th\'eor\`eme~\ref{thm:gg}, la condition suivante pour le pas de temps~:
+\begin{equation}
+  \Delta t < \dfrac{2}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
+\end{equation}
+
+
+ \item Le sch\'ema de Runge Kutta (ordre 3)
+Pour Runge-Kutta d'ordre 3, on a~:
+\begin{equation}
+ \omega^{n+1} = \bigl(1 + \nabla u.\Delta t + \frac{1}{2}(\nabla u.\Delta t)^2 + \frac{1}{6}(\nabla u.\Delta t)^3 \bigr)\ \omega^{n}
+\end{equation}
+\begin{figure}[ht]
+\centering
+ \includegraphics[width=0.6\linewidth]{RK3}
+\caption{Repr\'esentation graphique des \'equations $y=|1+x+ \frac{1}{2}x^2+ \frac{1}{6}x^3|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta  t$}
+\end{figure}
+
+On obtient alors en appliquant aussi le th\'eor\`eme~\ref{thm:gg}, la condition suivante pour le pas de temps~:
+\begin{equation}
+  \Delta t < \dfrac{2.5127}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
+\end{equation}
+
+ \item Le sch\'ema de Runge Kutta (ordre 4)
+Pour Runge-Kutta d'ordre 4, on a~:
+\begin{equation}
+ \omega^{n+1} = \bigl(1 + \nabla u.\Delta t + \frac{1}{2}(\nabla u.\Delta t)^2 + \frac{1}{6}(\nabla u.\Delta t)^3 + \frac{1}{24}(\nabla u.\Delta)^4 \bigr)\ \omega^{n}
+\end{equation}
+\begin{figure}[ht]
+\centering
+ \includegraphics[width=0.6\linewidth]{RK4}
+\caption{Repr\'esentation graphique des \'equations $y=|1+x+ \frac{1}{2}x^2+ \frac{1}{6}x^3+ \frac{1}{24}x^4|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta  t$}
+\end{figure}
+
+On obtient alors en appliquant aussi le th\'eor\`eme~\ref{thm:gg}, la condition suivante pour le pas de temps~:
+\begin{equation}
+  \Delta t < \dfrac{2.7853}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
+\end{equation}
+\end{itemize}
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|c|c|c|}
+\hline
+Sch\'ema & $C_\text{Stretching}$ & Ordre \\
+\hline
+Euler & 2 &1 \\
+\hline
+Runge-Kutta 2($\alpha =\frac12$) & 2 & 2\\
+\hline
+Runge-Kutta 3 & 2.5127& 3 \\
+\hline
+Runge-Kutta 4 & 2.7853 & 4\\
+\hline
+\end{tabular}
+
+\caption{\label{tab:Cstretch} Tableau r\'ecapitulatif des constantes intervenants dans la condition de stabilit\'e du sch\'ema en temps}
+\end{table}
+
+
+Il faut maintenant comparer ces majorations \autoref{tab:Cstretch}, avec la condition de stabilit\'e sur l'advection qui est la suivante~:
+\begin{equation}
+ \Delta t \simeq \dfrac{C}{\max_i|\frac{\partial u_i}{\partial x_i}|_{L_{\infty}}}
+\end{equation}
+avec $C$ qui vaut en g\'en\'eral 1.
+
+\section{\'Etude sur un cas test}
+L'\'etude de l'influence de la discr\'etisation en temps a \'et\'e faite sur l'exemple suivant~:
+\begin{align}
+\label{eq:exemplestretch}
+ \overrightarrow{u} =
+\left(\begin{array}{l}
+2 \sin^2(\pi x) \sin (2 \pi y) \sin(2 \pi z) \cos(\pi t) \\
+- \sin(2 \pi x) \sin^2 (\pi y) \sin(2 \pi z) \cos(\pi t) \\
+- \sin(2\pi x) \sin (2 \pi y) \sin^2(\pi z) \cos(\pi t)
+\end{array}\right)&&\\
+ \overrightarrow{w} =
+\left(\begin{array}{l}
+2 \pi \cos(\pi t) \sin(2 \pi x) \biggl(- \cos(2 \pi y) \sin^2(\pi z) + \cos(2 \pi z)\sin^2(\pi y) \biggr) \\
+2 \pi \cos(\pi t) \sin(2 \pi y) \biggl(2 \cos(2 \pi z) \sin^2(\pi x) + \cos(2 \pi x)\sin^2(\pi z) \biggr) \\
+- 2 \pi \cos(\pi t) \sin(2 \pi z) \biggl( \cos(2 \pi x) \sin^2(\pi y) + \cos(2 \pi y)\sin^2(\pi x) \biggr) 
+\end{array}\right)&&\\
+\text{div} (\overrightarrow{u}) = 0 &&
+\end{align}
+La vorticit\'e est initialis'ee en prenant le rotationnel de u~:
+\begin{equation}
+\overrightarrow{w}= \overrightarrow{\text{rot}}u 
+\end{equation}
+
+Pour cette exemple, et en prenant l'\'equation \autoref{eq:gradstretch} avec un sch\'ema aux diff\'erences finis centr\'es d'ordre 4 en espace avec un stencile de 5, on obtient les conditions de stabilit\'es suivantes pour le sch\'ema en temps \autoref{tab:stabexemple}.
+
+\begin{table}[ht]
+ \centering
+\begin{tabular}{|c|c|c|}
+\hline
+\multicolumn{3}{|c|}{$\Delta t_\text{advection} = 0.44331613$}  \\
+\hline
+\hline
+Sch\'ema & $\max(\Delta t_\text{Stretching})$ & Ordre \\
+\hline
+Euler & 0.14994711& 1\\
+\hline
+Runge-Kutta 2 &0.14994711 & 2\\
+Runge-Kutta 3 &0.18838605 & 3\\
+Runge-Kutta 4 &0.20882384 & 4\\
+\hline
+Adams-Moulton 3 & 0.44984132 & 3 \\
+\hline
+\end{tabular}
+\caption{\label{tab:stabexemple} Conditions de stabilit\'e sur le pas de temps pour l'exemple \eqref{eq:exemplestretch}}
+\end{table}
+
+
+
+\section{Validation Sch\'ema Runge-Kutta}
+Pour valider les sch\'emas de Runge-Kutta, nous sommes partis de l'exemple suivant~:
+\begin{align}
+ \overrightarrow{u} =
+\left(\begin{array}{l}
+\sin(x)\\
+\sin(y)\\
+\sin(z)
+\end{array}\right)&&\\
+ \overrightarrow{w} =
+\left(\begin{array}{l}
+1+t\\
+1+t\\
+1+t
+\end{array}\right)&&
+\end{align}
+
+La solution analytique est alors~:
+\begin{equation}
+ \omega_\text{analytique} = 
+\left(\begin{array}{l}
+(\frac{t^2}{2} + t) \cos(x)\\
+(\frac{t^2}{2} + t) \cos(y)\\
+(\frac{t^2}{2} + t) \cos(z)
+\end{array}\right)
+\end{equation}
+
+Seulement cette solution analytique n'est valable que pour le sch\'ema d'Euler. Pour les sch\'emas de type Runge-Kutta, il faut rajouter un terme correcteur d\^u aux erreurs de d\'erivation aux diff\'erents temps.\\
+On a alors~:
+\begin{equation}
+ \omega =  \omega_\text{analytique} + F(t)
+\end{equation}
+
+Les fonctions correctrices pour $t^1$ pour cet exemple sont les suivantes~:
+\begin{tabular}{|c|c|}
+\hline
+Sch\'ema & Terme correcteur\\
+ \hline
+RK 2 & $\left(\begin{array}{l}
+\cos^2(x)(\frac{\Delta t^2}{2} + \frac{\Delta t^3}{4})\\
+\cos^2(y)(\frac{\Delta t^2}{2} + \frac{\Delta t^3}{4})\\
+\cos^2(z)(\frac{\Delta t^2}{2} + \frac{\Delta t^3}{4})
+\end{array}\right)
+$\\ \hline
+RK 3 & $\left(\begin{array}{l}
+\cos^2(x)(\frac{\Delta t^2}{2} + \frac{\Delta t^3}{2})+\cos^3(x)(\frac{\Delta t^3}{6} + \frac{\Delta t^4}{6})\\
+\cos^2(y)(\frac{\Delta t^2}{2} + \frac{\Delta t^3}{2})+\cos^3(y)(\frac{\Delta t^3}{6} + \frac{\Delta t^4}{6})\\
+\cos^2(z)(\frac{\Delta t^2}{2} + \frac{\Delta t^3}{2})+\cos^3(z)(\frac{\Delta t^3}{6} + \frac{\Delta t^4}{6})     
+\end{array}\right)
+$\\ \hline
+RK 4 & $\left(\begin{array}{l}
+\frac{1}{6} \cos(x)\biggl(-\Delta t^2 + \cos(x)\Delta t^2(3 + \frac{7}{2}\Delta t)\\ + \frac{\Delta t^3}{2}\cos^2(x)(2+3\Delta t) + \frac{\Delta t^4}{4} \cos^3(x)(1 + 2\Delta t)\biggr)\\
+\frac{1}{6} \cos(y)\biggl(-\Delta t^2 + \cos(y)\Delta t^2(3 + \frac{7}{2}\Delta t)\\ + \frac{\Delta t^3}{2}\cos^2(y)(2+3\Delta t) + \frac{\Delta t^4}{4} \cos^3(y)(1 + 2\Delta t)\biggr)\\
+\frac{1}{6} \cos(z)\biggl(-\Delta t^2 + \cos(z)\Delta t^2(3 + \frac{7}{2}\Delta t) \\+ \frac{\Delta t^3}{2}\cos^2(z)(2+3\Delta t) + \frac{\Delta t^4}{4} \cos^3(z)(1 + 2\Delta t)\biggr)
+\end{array}\right)
+$\\ \hline
+\end{tabular}
+
+
+Avec les termes correcteurs nous obtenons les erreurs suivantes figures(\ref{fig:errorRK2}-\ref{fig:errorRK3}-\ref{fig:errorRK3TVD}-\ref{fig:errorRK4}).
+\begin{figure}[ht]
+ \centering
+\includegraphics[width=1\linewidth]{RK2_error}
+\caption{\label{fig:errorRK2} \`A gauche la diff\'erencre entre la solution num\'erique pour le sch\'ema de Runge-Kutta 2 et analytique sans terme correctif, \`a droite avec terme correctif.}
+\end{figure}
+
+\begin{figure}[ht]
+ \centering
+\includegraphics[width=1\linewidth]{RK3_error}
+\caption{\label{fig:errorRK3} \`A gauche la diff\'erencre entre la solution num\'erique pour le sch\'ema de Runge-Kutta 3 et analytique sans terme correctif, \`a droite avec terme correctif.}
+\end{figure}
+\begin{figure}[ht]
+ \centering
+\includegraphics[width=1\linewidth]{RK3_TVD_error}
+\caption{\label{fig:errorRK3TVD} \`A gauche la diff\'erencre entre la solution num\'erique pour le sch\'ema de Runge-Kutta 3 TVD et analytique sans terme correctif, \`a droite avec terme correctif.}
+\end{figure}
+\begin{figure}[ht]
+ \centering
+\includegraphics[width=1\linewidth]{RK4_error}
+\caption{\label{fig:errorRK4} \`A gauche la diff\'erencre entre la solution num\'erique pour le sch\'ema de Runge-Kutta 4 et analytique sans terme correctif, \`a droite avec terme correctif.}
+\end{figure}
+
+Les diff\'erents sch\'emas de Runge-Kutta 3 que l'on a impl\'ement\'e~:
+\begin{itemize}
+ \item[Version 1] 
+$\begin{array}{ll}
+\omega^{n+1} = \omega^n + \frac{1}{4}(K_1 + 3K_3)\\
+K_1= \Delta t f(u^n,\omega^n,t^n)\\
+K_2= \Delta t f(u^n,\omega^n + \frac{K_1}{3},t^n + \frac{1}{3}\Delta t)\\
+K_3= \Delta t f(u^n,\omega^n+ \frac{2K_2}{3},t^n+ \frac{2}{3}\Delta t)
+\end{array}
+$
+ \item[Version 2] 
+$\begin{array}{ll}
+\omega^{n+1} = \omega^n + \frac{1}{6}(K_1 + 4K_2+ K_3)\\
+K_1= \Delta t f(u^n,\omega^n,t^n)\\
+K_2= \Delta t f(u^n,\omega^n + \frac{K_1}{2},t^n + \frac{1}{2}\Delta t)\\
+K_3= \Delta t f(u^n,\omega^n +2K_2 -K_1,t^n + \Delta t)
+\end{array}
+$
+ \item[Version TVD] 
+$\begin{array}{ll}
+\omega^{n+1} = \omega^n + \frac{2}{3}(K_2 + K_3)\\
+K_1= \Delta t f(u^n,\omega^n,t^n)\\
+K_2= \Delta t f(u^n,\frac{3}{4}\omega^n + \frac{K_1}{4},t^n + \Delta t)\\
+K_3= \Delta t f(u^n,\omega^n + K_2,t^n + \frac{1}{2}\Delta t)
+\end{array}
+$
+\end{itemize}
+
+\end{document}
diff --git a/Docs/images/Euler.png b/Docs/images/Euler.png
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diff --git a/HySoP/hysop/operator/differentialOperator.py b/HySoP/hysop/operator/differentialOperator.py
index db3e5f715..e43371744 100755
--- a/HySoP/hysop/operator/differentialOperator.py
+++ b/HySoP/hysop/operator/differentialOperator.py
@@ -27,7 +27,7 @@ class DifferentialOperator(ContinuousOperator):
         self.choice = choice
         self.addVariable([field1, field2])
 
-    def discretize(self, field1=None, field2=None, result=None): # changer commentaires
+    def discretize(self, field1=None, field2=None, result=None, maxgersh=None): # changer commentaires
         """
         Stretching operator discretization method.
         Create a discrete Stretching operator from given specifications.
@@ -36,9 +36,10 @@ class DifferentialOperator(ContinuousOperator):
         @param idVorticityD : Index of vorticity discretisation to use.
         @param res_vorticity : result stretch (vorticity).
         @param method : the method to use.
+        @param maxgersh : result Gershgorin's condition.
         """
         if self.discreteOperator is None:
-            self.discreteOperator = DifferentialOperator_d(self, field1, field2, result, self.choice)
+            self.discreteOperator = DifferentialOperator_d(self, field1, field2, result, self.choice, maxgersh)
 
     def __str__(self):
         """ToString method"""
diff --git a/HySoP/hysop/operator/differentialOperator_d.py b/HySoP/hysop/operator/differentialOperator_d.py
index 992352b0b..9493292fb 100755
--- a/HySoP/hysop/operator/differentialOperator_d.py
+++ b/HySoP/hysop/operator/differentialOperator_d.py
@@ -9,14 +9,13 @@ import numpy as np
 import sys
 import time
 
-
 class DifferentialOperator_d(DiscreteOperator):
     """
     Operator representation.
     DiscreteOperator.DiscreteOperator specialization.
     """
 
-    def __init__(self, opDif, field1, field2, result=None, choice=None):
+    def __init__(self, opDif, field1, field2, result=None, choice=None, maxgersh=None):
         """
         Constructor.
         Create a Stretching operator on a given continuous domain.
@@ -28,6 +27,7 @@ class DifferentialOperator_d(DiscreteOperator):
         self.field1 = field1
         self.field2 = field2
         self.result = result
+        self.maxgersh = maxgersh
         self.choice = choice
         self.compute_time = 0.
 
@@ -43,9 +43,9 @@ class DifferentialOperator_d(DiscreteOperator):
             resolution = self.field2.topology.mesh.resolution
             ind=np.arange(resolution[0])
 
-        if self.choice == 'divergenceProduct':
+        if self.choice == 'masseConservation':
 ##           second order scheme
-##           X components of aux and result
+##           X components of temp and result
 #            temp1 = ( self.field1[0][np.roll(ind,-1),...] * self.field2[0][np.roll(ind,-1),...] -\
 #                     self.field1[0][np.roll(ind,+1),...] * self.field2[0][np.roll(ind,+1),...]) / (2. * mesh_size[0]) 
 
@@ -57,7 +57,7 @@ class DifferentialOperator_d(DiscreteOperator):
 
 #            self.result[0][...]=  temp1 + temp2 + temp3
 
-##           Y components of aux and result
+##           Y components of temp and result
 #            temp1 = (self.field1[0][np.roll(ind,-1),...] * self.field2[1][np.roll(ind,-1),...] -\
 #                     self.field1[0][np.roll(ind,+1),...] * self.field2[1][np.roll(ind,+1),...] ) / (2. * mesh_size[0]) 
 
@@ -69,7 +69,7 @@ class DifferentialOperator_d(DiscreteOperator):
 
 #            self.result[1][...]= temp1 + temp2 +temp3
 
-##           Z components of aux and result
+##           Z components of temp and result
 #            temp1 = (self.field1[0][np.roll(ind,-1),...] * self.field2[2][np.roll(ind,-1),...] -\
 #                     self.field1[0][np.roll(ind,+1),...] * self.field2[2][np.roll(ind,+1),...] ) / (2. * mesh_size[0]) 
 
@@ -80,9 +80,7 @@ class DifferentialOperator_d(DiscreteOperator):
 #                     self.field1[2][...,np.roll(ind,+1)] * self.field2[2][...,np.roll(ind,+1)]) / (2. * mesh_size[2]) 
 
 
-
-
-#           X components of aux and result
+#           X components of temp and result
             temp1 = (1.0 * self.field1[0][np.roll(ind,+2),...] * self.field2[0][np.roll(ind,+2),...] -
                      8.0 * self.field1[0][np.roll(ind,+1),...] * self.field2[0][np.roll(ind,+1),...] +\
                      8.0 * self.field1[0][np.roll(ind,-1),...] * self.field2[0][np.roll(ind,-1),...] -\
@@ -100,7 +98,7 @@ class DifferentialOperator_d(DiscreteOperator):
 
             self.result[0][...]=  temp1 + temp2 + temp3
 
-#           Y components of aux and result
+#           Y components of temp and result
             temp1 = (1.0 * self.field1[0][np.roll(ind,+2),...] * self.field2[1][np.roll(ind,+2),...] -
                      8.0 * self.field1[0][np.roll(ind,+1),...] * self.field2[1][np.roll(ind,+1),...] +\
                      8.0 * self.field1[0][np.roll(ind,-1),...] * self.field2[1][np.roll(ind,-1),...] -\
@@ -118,7 +116,7 @@ class DifferentialOperator_d(DiscreteOperator):
 
             self.result[1][...]= temp1 + temp2 +temp3
 
-#           Z components of aux and result
+#           Z components of temp and result
             temp1 = (1.0 * self.field1[0][np.roll(ind,+2),...] * self.field2[2][np.roll(ind,+2),...] -
                      8.0 * self.field1[0][np.roll(ind,+1),...] * self.field2[2][np.roll(ind,+1),...] +\
                      8.0 * self.field1[0][np.roll(ind,-1),...] * self.field2[2][np.roll(ind,-1),...] -\
@@ -135,9 +133,11 @@ class DifferentialOperator_d(DiscreteOperator):
                      1.0 * self.field1[2][...,np.roll(ind,-2)] * self.field2[2][...,np.roll(ind,-2)]) / (12. * mesh_size[2]) 
 
             self.result[2][...]= temp1 + temp2 +temp3
-        elif self.choice == 'divergenceProductF':
+
+
+        elif self.choice == 'masseConservationF':
             
-#           X components of aux and result
+#           X components of temp and result
             temp1 = (-25./12. * self.field1[0][np.roll(ind,0),...] * self.field2[0][np.roll(ind,0),...] +
                        4.     * self.field1[0][np.roll(ind,+1),...] * self.field2[0][np.roll(ind,+1),...] +\
                       -3.     * self.field1[0][np.roll(ind,+2),...] * self.field2[0][np.roll(ind,+2),...] +\
@@ -158,7 +158,7 @@ class DifferentialOperator_d(DiscreteOperator):
 
             self.result[0][...]=  temp1 + temp2 + temp3
 
-#           Y components of aux and result
+#           Y components of temp and result
             temp1 = (-25./12.  * self.field1[0][np.roll(ind,+0),...] * self.field2[1][np.roll(ind,+0),...] +
                        4.      * self.field1[0][np.roll(ind,+1),...] * self.field2[1][np.roll(ind,+1),...] +\
                       -3.      * self.field1[0][np.roll(ind,+2),...] * self.field2[1][np.roll(ind,+2),...] +\
@@ -179,7 +179,7 @@ class DifferentialOperator_d(DiscreteOperator):
 
             self.result[1][...]= temp1 + temp2 +temp3
 
-#           Z components of aux and result
+#           Z components of temp and result
             temp1 = (-25./12.  * self.field1[0][np.roll(ind,+0),...] * self.field2[2][np.roll(ind,+0),...] +
                        4.      * self.field1[0][np.roll(ind,+1),...] * self.field2[2][np.roll(ind,+1),...] +\
                       -3.      * self.field1[0][np.roll(ind,+2),...] * self.field2[2][np.roll(ind,+2),...] +\
@@ -200,8 +200,73 @@ class DifferentialOperator_d(DiscreteOperator):
 
             self.result[2][...]= temp1 + temp2 +temp3
 
+        elif self.choice == 'divConservation':
+
+##           fourth order scheme
+#           X components of temp and result
+            temp1 = (1.0 * self.field2[0][np.roll(ind,+2),...] -
+                     8.0 * self.field2[0][np.roll(ind,+1),...] +\
+                     8.0 * self.field2[0][np.roll(ind,-1),...] -\
+                     1.0 * self.field2[0][np.roll(ind,-2),...]) / (12. * mesh_size[0]) 
+
+            temp2 = (1.0 * self.field2[0][:,np.roll(ind,+2),:] -\
+                     8.0 * self.field2[0][:,np.roll(ind,+1),:] +\
+                     8.0 * self.field2[0][:,np.roll(ind,-1),:] -\
+                     1.0 * self.field2[0][:,np.roll(ind,-2),:]) / (12. * mesh_size[1]) 
+                                                       
+            temp3 = (1.0 * self.field2[0][...,np.roll(ind,+2)] -\
+                     8.0 * self.field2[0][...,np.roll(ind,+1)] +\
+                     8.0 * self.field2[0][...,np.roll(ind,-1)] -\
+                     1.0 * self.field2[0][...,np.roll(ind,-2)]) / (12. * mesh_size[2]) 
+
+            max1= np.max(abs(temp1)+abs(temp2)+abs(temp3))
+            maxadv1= np.max(abs(temp1))
+            self.result[0][...]=  temp1 * self.field1[0][...] + temp2 * self.field1[1][...] + temp3 * self.field1[2][...]
+
+#           Y components of temp and result
+            temp1 = (1.0 * self.field2[1][np.roll(ind,+2),...] -
+                     8.0 * self.field2[1][np.roll(ind,+1),...] +\
+                     8.0 * self.field2[1][np.roll(ind,-1),...] -\
+                     1.0 * self.field2[1][np.roll(ind,-2),...]) / (12. * mesh_size[0]) 
+
+            temp2 = (1.0 * self.field2[1][:,np.roll(ind,+2),:] -\
+                     8.0 * self.field2[1][:,np.roll(ind,+1),:] +\
+                     8.0 * self.field2[1][:,np.roll(ind,-1),:] -\
+                     1.0 * self.field2[1][:,np.roll(ind,-2),:]) / (12. * mesh_size[1]) 
+                                                       
+            temp3 = (1.0 * self.field2[1][...,np.roll(ind,+2)] -\
+                     8.0 * self.field2[1][...,np.roll(ind,+1)] +\
+                     8.0 * self.field2[1][...,np.roll(ind,-1)] -\
+                     1.0 * self.field2[1][...,np.roll(ind,-2)]) / (12. * mesh_size[2]) 
+
+            max2= np.max(abs(temp1)+abs(temp2)+abs(temp3))
+            maxadv2= np.max(abs(temp2))
+            self.result[1][...]= temp1 * self.field1[0][...] + temp2 * self.field1[1][...] + temp3 * self.field1[2][...]
+
+#           Z components of temp and result
+            temp1 = (1.0 * self.field2[2][np.roll(ind,+2),...] -
+                     8.0 * self.field2[2][np.roll(ind,+1),...] +\
+                     8.0 * self.field2[2][np.roll(ind,-1),...] -\
+                     1.0 * self.field2[2][np.roll(ind,-2),...]) / (12. * mesh_size[0]) 
+
+            temp2 = (1.0 * self.field2[2][:,np.roll(ind,+2),:] -\
+                     8.0 * self.field2[2][:,np.roll(ind,+1),:] +\
+                     8.0 * self.field2[2][:,np.roll(ind,-1),:] -\
+                     1.0 * self.field2[2][:,np.roll(ind,-2),:]) / (12. * mesh_size[1]) 
+                                                       
+            temp3 = (1.0 * self.field2[2][...,np.roll(ind,+2)] -\
+                     8.0 * self.field2[2][...,np.roll(ind,+1)] +\
+                     8.0 * self.field2[2][...,np.roll(ind,-1)] -\
+                     1.0 * self.field2[2][...,np.roll(ind,-2)]) / (12. * mesh_size[2]) 
+
+            max3= np.max(abs(temp1)+abs(temp2)+abs(temp3))
+            maxadv3= np.max(abs(temp3))
+            self.result[2][...]= temp1 * self.field1[0][...] + temp2 * self.field1[1][...] + temp3 * self.field1[2][...]
+            self.maxgersh[0] = max(max1,max2,max3)
+#            self.maxgersh[0] = max(maxadv1,maxadv2,maxadv3)
 
         elif self.choice == 'curl':
+
             temp1 = (1.0 * self.field1[2][:,np.roll(ind,+2),:] -
                      8.0 * self.field1[2][:,np.roll(ind,+1),:] +\
                      8.0 * self.field1[2][:,np.roll(ind,-1),:] -\
diff --git a/HySoP/hysop/operator/stretching_d.py b/HySoP/hysop/operator/stretching_d.py
index 1baec3f23..88ed5c5f1 100755
--- a/HySoP/hysop/operator/stretching_d.py
+++ b/HySoP/hysop/operator/stretching_d.py
@@ -9,6 +9,7 @@ from .differentialOperator import DifferentialOperator
 from .differentialOperator_d import DifferentialOperator_d
 from ..particular_solvers.integrator.euler import Euler
 from ..particular_solvers.integrator.runge_kutta2stretching import RK2Stretch
+from ..particular_solvers.integrator.runge_kutta3stretching import RK3Stretch
 from ..particular_solvers.integrator.runge_kutta4stretching import RK4Stretch
 from ..particular_solvers.integrator.integrator import ODESolver
 import numpy as np
@@ -45,6 +46,7 @@ class Stretching_d(DiscreteOperator):
         ## output fields
         self.output = [self.res_vorticity] 
         self.method = method
+        self.maxgersh = np.zeros(1, dtype=PARMES_REAL, order=ORDER)
         ## self.name = "stretching"
 
     def apply(self, t, dt):
@@ -61,38 +63,73 @@ class Stretching_d(DiscreteOperator):
          -integrate with the chosen method (RK4. RK2. euler)
         """
         self.compute_time = time.time()
+        
 
         if self.method is not None:
             ## Space discretisation of div (wu)
-            self.stretchTerm = DifferentialOperator(self.stretch.vorticity, self.stretch.velocity, choice='divergenceProduct')
-            self.stretchTerm.discretize(self.vorticity, self.velocity, self.stretchTermResult)
+            self.stretchTerm = DifferentialOperator(self.stretch.vorticity, self.stretch.velocity, choice='divConservation')
+            self.stretchTerm.discretize(self.vorticity, self.velocity, self.stretchTermResult, self.maxgersh)
             self.stretchTerm.discreteOperator.apply()
 
             # Time integration of dw/dt = div(wu)
 #            self.method=Euler
-            if Euler == self.method :
-                print "Euler"
-                methodInt=self.method(self.stretchTerm.discreteOperator)
-                for i in range(self.vorticity.topology.dim):
-                    self.res_vorticity[i][...] = methodInt.integrate(self.vorticity, self.stretchTermResult, t, dt, direction=i) 
-
-#            self.method=RK2Stretch
-            if RK2Stretch == self.method :
-                print "RK2"
-                methodInt=self.method(self.stretchTerm.discreteOperator)
-                self.res_vorticity = methodInt.integrate(self.stretchTerm, self.stretchTermResult, self.vorticity, self.velocity, self.res_vorticity, t, dt, self.vorticity.topology.dim)
-
-#            self.method=RK4Stretch
-            if RK4Stretch == self.method :
-                print "RK4"
-                methodInt=self.method(self.stretchTerm.discreteOperator)
-                self.res_vorticity = methodInt.integrate(self.stretchTerm, self.stretchTermResult, self.vorticity, self.velocity, self.res_vorticity, t, dt, self.vorticity.topology.dim) 
+
+            ndt = self.stabilityTest(dt, self.maxgersh, self.method)
+            dtstretch = dt/float(ndt)
+
+            for i in range (ndt) :
+                if Euler == self.method :
+                    print "Euler"
+                    methodInt=self.method(self.stretchTerm.discreteOperator)
+                    for i in range(self.vorticity.topology.dim):
+                        self.res_vorticity[i][...] = methodInt.integrate(self.vorticity, self.stretchTermResult, t, dtstretch, direction=i) 
+
+#               self.method=RK2Stretch
+                if RK2Stretch == self.method :
+                    print "RK2"
+                    methodInt=self.method(self.stretchTerm.discreteOperator)
+                    self.res_vorticity = methodInt.integrate(self.stretchTerm, self.stretchTermResult, self.vorticity, self.velocity, self.res_vorticity, t, dtstretch, self.vorticity.topology.dim)
+
+#               self.method=RK3Stretch
+                if RK3Stretch == self.method :
+                    print "RK3"
+                    methodInt=self.method(self.stretchTerm.discreteOperator)
+                    self.res_vorticity = methodInt.integrate(self.stretchTerm, self.stretchTermResult, self.vorticity, self.velocity, self.res_vorticity, t, dtstretch, self.vorticity.topology.dim)
+
+
+#                self.method=RK4Stretch
+                if RK4Stretch == self.method :
+                    print "RK4"
+                    methodInt=self.method(self.stretchTerm.discreteOperator)
+                    self.res_vorticity = methodInt.integrate(self.stretchTerm, self.stretchTermResult, self.vorticity, self.velocity, self.res_vorticity, t, dtstretch, self.vorticity.topology.dim) 
 
             self.vorticity.data= np.copy(self.res_vorticity.data)
             self.compute_time = time.time() - self.compute_time
             self.total_time += self.compute_time
         return self.compute_time
 
+    def stabilityTest(self, dt, maxi, method):
+        dtstretch = dt
+        cststretch =0.
+        if method == Euler :
+            cststretch = 2.0
+            dtstretch = min(dtstretch, cststretch/maxi)
+        if method == RK2Stretch :
+            cststretch = 2.0
+            dtstretch = min(dtstretch, cststretch/maxi)
+#        if method == RK3Stretch :
+#            cststretch = 2.5127
+#            dtstretch = min(dtstretch, cststretch/maxi)
+        if method == RK4Stretch :
+            cststretch = 2.7853
+            dtstretch = min(dtstretch, cststretch/maxi)
+        if dt == dtstretch :
+            print "dt, ndt , dtstretch, upperbound", dt, int(dt/dtstretch), dt/float(int(dt/dtstretch)), 6./maxi
+            return int(dt/dtstretch)
+        else :
+            print "dt, ndt , dtstretch, upperbound", dt, int(dt/dtstretch)+1, dt/float(int(dt/dtstretch)+1), cststretch/maxi
+            return int(dt/dtstretch) +1
+
     def printComputeTime(self):
         self.timings_info[0] = "\"Stretching total\""
         self.timings_info[1] = str(self.total_time)
diff --git a/HySoP/hysop/particular_solvers/basic.py b/HySoP/hysop/particular_solvers/basic.py
index 559b0146d..c7300e269 100644
--- a/HySoP/hysop/particular_solvers/basic.py
+++ b/HySoP/hysop/particular_solvers/basic.py
@@ -7,6 +7,8 @@ from solver import Solver
 #from integrator.runge_kutta4 import RK4
 from .integrator.integrator import ODESolver
 from .integrator.euler import Euler
+from .integrator.runge_kutta2stretching import RK2Stretch
+from .integrator.runge_kutta4stretching import RK4Stretch
 from .integrator.runge_kutta2 import RK2
 from .integrator.runge_kutta4 import RK4
 from ..fields.continuous import ContinuousField
diff --git a/HySoP/hysop/particular_solvers/integrator/runge_kutta2stretching.py b/HySoP/hysop/particular_solvers/integrator/runge_kutta2stretching.py
index cddb82159..8fb0bca80 100755
--- a/HySoP/hysop/particular_solvers/integrator/runge_kutta2stretching.py
+++ b/HySoP/hysop/particular_solvers/integrator/runge_kutta2stretching.py
@@ -44,12 +44,13 @@ class RK2Stretch(ODESolver):
         @param t : current time.
         @param dt : time step.
         """
+        maxgersh = np.zeros(1, dtype=PARMES_REAL, order=ORDER)
         resultTmp = np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
         K1=np.asarray(fd)
         K1[:,:,:,:]= K1[:,:,:,:] *0.5*dt + field1[:,:,:] 
         if False in [(field1[i][...].shape == field2[i][...].shape) for i in xrange(dimension)] :
             print "Error, not the same mesh on field1 and  on field2"
-        f.discreteOperator = DifferentialOperator_d(opDif=f, field1=K1, field2=field2, result=resultTmp, choice='divergenceProduct')
+        f.discreteOperator = DifferentialOperator_d(opDif=f, field1=K1, field2=field2, result=resultTmp, choice='divConservation', maxgersh=maxgersh)
         f.discreteOperator.apply()
         res_vorticity[:,:,:] = field1[:,:,:] + resultTmp[:,:,:,:] * dt
         return res_vorticity
diff --git a/HySoP/hysop/particular_solvers/integrator/runge_kutta3stretching.py b/HySoP/hysop/particular_solvers/integrator/runge_kutta3stretching.py
new file mode 100644
index 000000000..789659b3c
--- /dev/null
+++ b/HySoP/hysop/particular_solvers/integrator/runge_kutta3stretching.py
@@ -0,0 +1,84 @@
+# -*- coding: utf-8 -*-
+"""
+@package Utils
+RK3 method interface.
+"""
+from .integrator import ODESolver
+from ...operator.differentialOperator_d import DifferentialOperator_d
+from ...operator.differentialOperator import DifferentialOperator
+from parmepy.constants import *
+import numpy as np
+import copy
+
+class RK3Stretch(ODESolver):
+    """
+    ODESolver implementation for solving an equation system with RK3 method.
+    y'(t) = f(t,y)
+
+    y(t_n+1)= y(t_n) + dt*y'[y(t_n)+dt/2*y'(y(t_n))].
+
+    """
+    def __init__(self, f=None, dim=3):
+        """
+        Constructor.
+
+        @param f function f(t,y) : Right hand side of the equation to solve.
+        @param dim : dimensions
+        @param conditions : function to apply boundary conditions.
+        """
+        ODESolver.__init__(self, f)
+
+
+    def integrate(self, f, fd, field1, field2, res_vorticity, t, dt, dimension):
+        """
+        Integration step for RK3 Method
+         yp= y + dt*y'[y+dt/2*y'(y)].
+
+        @param y : position at time t.
+        @param f : operator to calcul y'(y) value at time t.
+        @param fd : result position at time t + dt
+        @param field1 : vorticity field discretized
+        @param field2 : velocity field discretized
+        @param t : current time.
+        @param dt : time step.
+        """
+        # RK3 TVD
+        maxgersh = np.zeros(1, dtype=PARMES_REAL, order=ORDER)
+        K1=np.asarray(fd)
+        K2=np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
+        K1[:,:,:,:] = field1[:,:,:] + dt * K1[:,:,:,:]
+        f.discreteOperator = DifferentialOperator_d(f, K1, field2, K2, choice='divConservation',maxgersh=maxgersh)
+        f.discreteOperator.apply()
+        K2[:,:,:,:] = 3./4. * field1[:,:,:] + 1./4. * (K1[:,:,:,:] + dt * K2[:,:,:,:])
+        f.discreteOperator = DifferentialOperator_d(f, K2, field2, res_vorticity, choice='divConservation',maxgersh=maxgersh)
+        f.discreteOperator.apply()
+        res_vorticity[:,:,:] = 1./3. * field1[:,:,:] + 2./3. * (K2[:,:,:,:] + dt * res_vorticity[:,:,:])
+        return res_vorticity
+
+        # RK3 V1
+#        maxgersh = np.zeros(1, dtype=PARMES_REAL, order=ORDER)
+#        K1=np.asarray(fd)
+#        K2=np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
+#        K3=np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
+#        resultTmp = np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
+#        K1[:,:,:,:] = dt * K1[:,:,:,:]
+#        resultTmp[:,:,:] = field1[:,:,:] + 1./3. * K1[:,:,:,:]
+#        f.discreteOperator = DifferentialOperator_d(f, resultTmp, field2, K2, choice='divConservation',maxgersh=maxgersh)
+#        f.discreteOperator.apply()
+#        K2[:,:,:,:] = field1[:,:,:] + 2./3. * dt * K2[:,:,:,:]
+#        f.discreteOperator = DifferentialOperator_d(f, K2, field2, K3, choice='divConservation',maxgersh=maxgersh)
+#        f.discreteOperator.apply()
+#        K3[:,:,:,:] = dt * K3[:,:,:,:]
+#        res_vorticity[:,:,:] = field1[:,:,:] + 1./4. * (K1[:,:,:,:] + 3 * K3[:,:,:,:])
+#        return res_vorticity
+
+
+    def __str__(self):
+        """ToString method"""
+        return "RK3 Method"
+
+
+if __name__ == "__main__":
+    print __doc__
+    print "- Provided class : RK3"
+    print RK3.__doc__
diff --git a/HySoP/hysop/particular_solvers/integrator/runge_kutta4stretching.py b/HySoP/hysop/particular_solvers/integrator/runge_kutta4stretching.py
index 3a3d05309..6f35ba8f2 100755
--- a/HySoP/hysop/particular_solvers/integrator/runge_kutta4stretching.py
+++ b/HySoP/hysop/particular_solvers/integrator/runge_kutta4stretching.py
@@ -42,6 +42,7 @@ class RK4Stretch(ODESolver):
         @param t : current time.
         @param dt : time step.
         """
+        maxgersh = np.zeros(1, dtype=PARMES_REAL, order=ORDER)
         resultTmp = np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
         K1=np.asarray(fd)
         K2=np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
@@ -49,15 +50,15 @@ class RK4Stretch(ODESolver):
         K4=np.asarray([np.zeros((field2.topology.resolution), dtype=PARMES_REAL, order=ORDER) for d in xrange(field2.topology.dim)])
         K1[:,:,:,:] =K1[:,:,:,:]* dt
         resultTmp[:,:,:,:] = K1[:,:,:,:] * 0.5 + field1[:,:,:]
-        f.discreteOperator = DifferentialOperator_d(f,resultTmp , field2, K2, choice='divergenceProduct')
+        f.discreteOperator = DifferentialOperator_d(f,resultTmp , field2, K2, choice='divConservation',maxgersh=maxgersh)
         f.discreteOperator.apply()
         K2[:,:,:,:] = K2[:,:,:,:] * dt
         resultTmp[:,:,:,:] = K2[:,:,:,:] * 0.5 + field1[:,:,:]
-        f.discreteOperator = DifferentialOperator_d(f, resultTmp, field2, K3, choice='divergenceProduct')
+        f.discreteOperator = DifferentialOperator_d(f, resultTmp, field2, K3, choice='divConservation',maxgersh=maxgersh)
         f.discreteOperator.apply()
         K3[:,:,:,:] = K3[:,:,:,:] * dt
         resultTmp[:,:,:,:] = K3[:,:,:,:] + field1[:,:,:]
-        f.discreteOperator = DifferentialOperator_d(f, resultTmp, field2, K4, choice='divergenceProduct')
+        f.discreteOperator = DifferentialOperator_d(f, resultTmp, field2, K4, choice='divConservation',maxgersh=maxgersh)
         f.discreteOperator.apply()
         K4[:,:,:,:] = K4[:,:,:,:] * dt
         res_vorticity[:,:,:] = field1[:,:,:] + 1./6. * (K1[:,:,:,:] + 2.0 * K2[:,:,:,:] + 2.0  * K3[:,:,:,:] + K4[:,:,:,:])
diff --git a/HySoP/hysop/test/test_operator/test_DivProduct.py b/HySoP/hysop/test/test_operator/test_DivProduct.py
index 4b5b63694..3581f8258 100755
--- a/HySoP/hysop/test/test_operator/test_DivProduct.py
+++ b/HySoP/hysop/test/test_operator/test_DivProduct.py
@@ -53,7 +53,7 @@ class test_DivProduct(unittest.TestCase):
         self.div.discreteOperator.apply()
         self.resol = self.topo3D.mesh.resolution
         self.FinalTime = 0.2
-        self.anal=self.analyticalDivProduct(self.topo3D.mesh.coords[0], \
+        self.anal=np.vectorize(self.analyticalDivProduct)(self.topo3D.mesh.coords[0], \
                                                      self.topo3D.mesh.coords[1], \
                                                      self.topo3D.mesh.coords[2])
         # Comparison with analytical solution 
@@ -75,9 +75,9 @@ class test_DivProduct(unittest.TestCase):
         return wx, wy, wz
 
     def analyticalDivProduct(self, x, y, z):
-        sx = np.cos(x)+ 0.*y*z
-        sy = np.cos(y)+ 0.*x*z
-        sz = np.cos(z)+ 0.*x*y
+        sx = np.cos(x)
+        sy = np.cos(y)
+        sz = np.cos(z)
         return sx, sy, sz
 
     def runTest(self):
diff --git a/HySoP/hysop/test/test_operator/test_Stretching.py b/HySoP/hysop/test/test_operator/test_Stretching.py
index 782077b48..6d8ee863f 100755
--- a/HySoP/hysop/test/test_operator/test_Stretching.py
+++ b/HySoP/hysop/test/test_operator/test_Stretching.py
@@ -1,6 +1,11 @@
 # -*- coding: utf-8 -*-
 import time
+from parmepy.particular_solvers.integrator.euler import Euler
+from parmepy.particular_solvers.integrator.runge_kutta2stretching import RK2Stretch
+from parmepy.particular_solvers.integrator.runge_kutta3stretching import RK3Stretch
+from parmepy.particular_solvers.integrator.runge_kutta4stretching import RK4Stretch
 import parmepy as pp
+import numpy as np
 from math import *
 import unittest
 
@@ -12,15 +17,35 @@ class test_Stretching(unittest.TestCase):
     """
 
     def vitesse(self,x, y, z):
-        vx = 1. + x
-        vy = - x * y
-        vz = x * y * z + 10.
+#        vx = 1. + x
+#        vy = - x * y
+#        vz = x * y * z + 10.
+        amodul = np.cos(np.pi*self.t/3)
+        pix = np.pi*x
+        piy = np.pi*y
+        piz = np.pi*z
+        pi2x = 2.*pix
+        pi2y = 2.*piy
+        pi2z = 2.*piz
+        vx = 2.*np.sin(pix)*np.sin(pix)*np.sin(pi2y)*np.sin(pi2z)*amodul
+        vy = -np.sin(pi2x)*np.sin(piy)*np.sin(piy)*np.sin(pi2z)*amodul
+        vz = -np.sin(pi2x)*np.sin(piz)*np.sin(piz)*np.sin(pi2y)*amodul
         return vx, vy, vz
 
     def vorticite(self,x, y, z):
-        wx = x * y
-        wy = y * z
-        wz = - y
+#        wx = x * y
+#        wy = y * z
+#        wz = - y
+        amodul = np.cos(np.pi*self.t/3)
+        pix = np.pi*x
+        piy = np.pi*y
+        piz = np.pi*z
+        pi2x = 2.*pix
+        pi2y = 2.*piy
+        pi2z = 2.*piz
+        wx = 2.* np.pi * np.sin(pi2x) * amodul*( - np.cos(pi2y)*np.sin(piz)*np.sin(piz)+ np.sin(piy)*np.sin(piy)*np.cos(pi2z) )
+        wy = 2.* np.pi * np.sin(pi2y) * amodul*( 2.*np.cos(pi2z)*np.sin(pix)*np.sin(pix)+ np.sin(piz)*np.sin(piz)*np.cos(pi2x) )
+        wz = -2.* np.pi * np.sin(pi2z) * amodul*( np.cos(pi2x)*np.sin(piy)*np.sin(piy)+ np.sin(pix)*np.sin(pix)*np.cos(pi2y) )
         return wx, wy, wz
 
     def scalaire(self,x, y, z):
@@ -33,9 +58,9 @@ class test_Stretching(unittest.TestCase):
     def testOperatorStretching(self):
         # Parameters
         nb = 32
-        timeStep = 0.02
-        finalTime = 0.06
-
+        timeStep = 0.09
+        finalTime = 0.09
+        self.t = 0.
         t0 = time.time()
 
         ## Domain
@@ -56,6 +81,7 @@ class test_Stretching(unittest.TestCase):
 
         ## Setting solver to Problem
         pb.setSolver(finalTime, timeStep, solver_type='basic')
+        pb.solver.ODESolver= RK3Stretch# RK2Stretch# Euler#RK4Stretch#
         pb.initSolver()
 
         t1 = time.time()
diff --git a/HySoP/hysop/test/test_particular_solvers/test_RK.py b/HySoP/hysop/test/test_particular_solvers/test_RK.py
index 63b15ce2b..7281ecb4e 100644
--- a/HySoP/hysop/test/test_particular_solvers/test_RK.py
+++ b/HySoP/hysop/test/test_particular_solvers/test_RK.py
@@ -3,6 +3,7 @@ import time
 import parmepy as pp
 from parmepy.particular_solvers.integrator.euler import Euler
 from parmepy.particular_solvers.integrator.runge_kutta2stretching import RK2Stretch
+from parmepy.particular_solvers.integrator.runge_kutta3stretching import RK3Stretch
 from parmepy.particular_solvers.integrator.runge_kutta4stretching import RK4Stretch
 from math import *
 import unittest
@@ -37,10 +38,31 @@ class test_Stretching(unittest.TestCase):
         sz = (0.5*(self.t*self.t)+self.t) * np.cos(z)
         return sx, sy, sz
 
+    def correctionRK2(self,x, y ,z):
+        cx = np.cos(x) * np.cos(x)* self.t*self.t*( ( 1./2. + self.t/4. ) )
+        cy = np.cos(y) * np.cos(y)* self.t*self.t*( ( 1./2. + self.t/4. ) )
+        cz = np.cos(z) * np.cos(z)* self.t*self.t*( ( 1./2. + self.t/4. ) )
+        return cx, cy, cz
+
+    def correctionRK3(self,x, y ,z):
+        cx = np.cos(x) * np.cos(x)* self.t*self.t*( ( 1./2. + self.t/2. ) + np.cos(x) *  self.t *(1./6. + self.t/6.) )
+        cy = np.cos(y) * np.cos(y)* self.t*self.t*( ( 1./2. + self.t/2. ) + np.cos(y) *  self.t *(1./6. + self.t/6.) )
+        cz = np.cos(z) * np.cos(z)* self.t*self.t*( ( 1./2. + self.t/2. ) + np.cos(z) *  self.t *(1./6. + self.t/6.) )
+        return cx, cy, cz
+
+    def correctionRK4(self,x, y ,z):
+        cx = 1./6. * (-np.cos(x) * (self.t)**2 + (np.cos(x))**2 * (self.t)**2 * (3. + 7./2. * self.t) + \
+                     1./2. * (self.t)**3 * (np.cos(x))**3 * (2. + 3. * self.t) + 1./4. * (self.t)**4 * (np.cos(x))**4 * (1. + 2. * self.t))
+        cy = 1./6. * (-np.cos(y) * (self.t)**2 + (np.cos(y))**2 * (self.t)**2 * (3. + 7./2. * self.t) + \
+                     1./2. * (self.t)**3 * (np.cos(y))**3 * (2. + 3. * self.t) + 1./4. * (self.t)**4 * (np.cos(y))**4 * (1. + 2. * self.t))
+        cz = 1./6. * (-np.cos(z) * (self.t)**2 + (np.cos(z))**2 * (self.t)**2 * (3. + 7./2. * self.t) + \
+                     1./2. * (self.t)**3 * (np.cos(z))**3 * (2. + 3. * self.t) + 1./4. * (self.t)**4 * (np.cos(z))**4 * (1. + 2. * self.t))
+        return cx, cy, cz
+
     def testOperatorStretching(self):
         # Parameters
-        nb = 32
-        timeStep = 0.0025
+        nb = 64
+        timeStep = 0.01
         finalTime = 0.01
 
         t0 = time.time()
@@ -70,8 +92,14 @@ class test_Stretching(unittest.TestCase):
         Analvorty[:,:,:,:]= np.vectorize(self.vorticite)(topo3D.mesh.coords[0], \
                                           topo3D.mesh.coords[1], \
                                           topo3D.mesh.coords[2])[:]
+        Analvorty2=np.asarray([np.zeros([nb, nb, nb], dtype=PARMES_REAL, order=ORDER) for d in xrange(3)])
+        Analvorty2[:,:,:,:]= np.vectorize(self.vorticite)(topo3D.mesh.coords[0], \
+                                          topo3D.mesh.coords[1], \
+                                          topo3D.mesh.coords[2])[:]
 
-        pb.solver.ODESolver=   RK2Stretch # Euler#RK4Stretch #
+        pb.solver.ODESolver= RK3Stretch #RK2Stretch #Euler#RK4Stretch #
+
+        print "\nODE SOLVER :: ", pb.solver.ODESolver
         pb.initSolver()
         t1 = time.time()
 
@@ -82,17 +110,32 @@ class test_Stretching(unittest.TestCase):
                                             topo3D.mesh.coords[1], \
                                             topo3D.mesh.coords[2])[:]
 
+        Analvorty2[:,:,:,:]=Analvorty2[:,:,:,:]+np.vectorize(self.analyticalSolution)(topo3D.mesh.coords[0], \
+                                            topo3D.mesh.coords[1], \
+                                            topo3D.mesh.coords[2])[:] + np.vectorize(self.correctionRK3)(topo3D.mesh.coords[0], \
+                                            topo3D.mesh.coords[1], \
+                                            topo3D.mesh.coords[2])[:]
+
+
 ## Visual comparison between the numerical resolution and the analytical resolution
         for i in range(1):
             plt.figure(1)
-            plt.subplot(211)
+            plt.subplot(221)
             plt.axis([-2*np.pi,2*np.pi,np.min(Analvorty[i][:,0,0]),np.max(Analvorty[i][:,0,0])])
             plt.plot(topo3D.mesh.coords[0][:,0,0], Analvorty[i][:,0,0], '-' ,topo3D.mesh.coords[0][:,0,0],vorti.discreteField[i].data[i][:,0,0], '-' )
             plt.legend([u"Solution Analytique", u"Solution Numérique"])
-            plt.subplot(212)
+            plt.subplot(223)
             plt.axis([-2*np.pi,2*np.pi,0,max(abs(Analvorty[i][:,0,0] - vorti.discreteField[i].data[i][:,0,0]))])
             plt.plot(topo3D.mesh.coords[0][:,0,0], abs(Analvorty[i][:,0,0] - vorti.discreteField[i].data[i][:,0,0]), '-' )
             plt.legend([u"Erreur"])
+            plt.subplot(222)
+            plt.axis([-2*np.pi,2*np.pi,np.min(Analvorty2[i][:,0,0]),np.max(Analvorty2[i][:,0,0])])
+            plt.plot(topo3D.mesh.coords[0][:,0,0], Analvorty2[i][:,0,0], '-' ,topo3D.mesh.coords[0][:,0,0],vorti.discreteField[i].data[i][:,0,0], '-' )
+            plt.legend([u"Solution Analytique", u"Solution Numérique"])
+            plt.subplot(224)
+            plt.axis([-2*np.pi,2*np.pi,0,max(abs(Analvorty2[i][:,0,0] - vorti.discreteField[i].data[i][:,0,0]))])
+            plt.plot(topo3D.mesh.coords[0][:,0,0], abs(Analvorty2[i][:,0,0] - vorti.discreteField[i].data[i][:,0,0]), '-' )
+            plt.legend([u"Erreur"])
             plt.show()
         npt.assert_array_less(abs(Analvorty[:,:,:,:] - vorti.discreteField[0].data[:,:,:]),0.002)
         tf = time.time()
-- 
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