Reorganize doc

docs/nstrophy_doc.tex

@@ -103,9 +103,85 @@ We truncate the Fourier modes and assume that $\hat u_k=0$ if $|k_1|>K_1$ or $|k
   \mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\}
   .
 \end{equation}
+\bigskip
 
-\subsubsection{Runge-Kutta methods}.
+{\bf Remark}:
-To solve the equation numerically, we will use Runge-Kutta methods, which compute an approximate value $\hat u_k^{(n)}$ for $\hat u_k(t_n)$.
+Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
+\begin{equation}
+  \hat u_{-k}=\hat u_k^*
+  .
+  \label{realu}
+\end{equation}
+Similarly,
+\begin{equation}
+  \hat g_{-k}=\hat g_k^*
+  .
+  \label{realg}
+\end{equation}
+Thus,
+\begin{equation}
+  T(\hat u,-k)
+  =
+  T(\hat u,k)^*
+  .
+  \label{realT}
+\end{equation}
+In order to keep the computation as quick as possible, we only compute and store the values for $k_1\geqslant 0$.
+(In fact, if we do not enforce the reality conditions, the computation has been found to be unstable.)
+
+\subsection{Reversible equation}
+\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}.
+It is defined directly in Fourier space:
+\begin{equation}
+  \partial_t\hat U_k=
+  -\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k
+  -i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
+  (q\cdot\hat U_p)\hat U_q
+  ,\quad
+  k\cdot\hat U_k=0
+\end{equation}
+where $\alpha$ is chosen such that the enstrophy is constant.
+In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}):
+\begin{equation}
+  \partial_t\hat u_k=
+  -\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k
+  +\hat g_k
+  +\frac{4\pi^2}{L^2|k|}T(\hat u,k)
+  .
+  \label{rns_k}
+\end{equation}
+To compute $\alpha$, we use the constancy of the enstrophy:
+\begin{equation}
+  \sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k
+  =0
+\end{equation}
+which, in terms of $\hat u$ is
+\begin{equation}
+  \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\partial_t\hat u_k
+  =0
+\end{equation}
+that is
+\begin{equation}
+  \frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4|\hat u_k|^2
+  =
+  \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\hat g_k
+  +\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_k^*T(\hat u,k)
+\end{equation}
+and so
+\begin{equation}
+  \alpha(\hat u)
+  =\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
+  .
+  \label{alpha}
+\end{equation}
+Note that, by\-~(\ref{realu})-(\ref{realT}),
+\begin{equation}
+  \alpha(\hat u)\in\mathbb R
+  .
+\end{equation}
+
+\subsection{Runge-Kutta methods}.
+To solve these equations numerically, we will use Runge-Kutta methods, which compute an approximate value $\hat u_k^{(n)}$ for $\hat u_k(t_n)$.
 {\tt nstrophy} supports the 4th order Runge-Kutta ({\tt RK4}) and 2nd order Runge-Kutta ({\tt RK2}) algorithms.
 In addition, several variable step methods are implemented:
 \begin{itemize}
@@ -190,30 +266,8 @@ It can be made by specifying the parameter {\tt adaptive\_cost}.
   These cost functions are selected by choosing {\tt adaptive\_cost=k3} and {\tt adaptive\_cost=k32} respectively.
 \end{itemize}
 
-\subsubsection{Reality}.
-Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
-\begin{equation}
-  \hat u_{-k}=\hat u_k^*
-  .
-  \label{realu}
-\end{equation}
-Similarly,
-\begin{equation}
-  \hat g_{-k}=\hat g_k^*
-  .
-  \label{realg}
-\end{equation}
-Thus,
-\begin{equation}
-  T(\hat u,-k)
-  =
-  T(\hat u,k)^*
-  .
-  \label{realT}
-\end{equation}
-In order to keep the computation as quick as possible, we only compute and store the values for $k_1\geqslant 0$.
 
-\subsubsection{FFT}. We compute T using a fast Fourier transform, defined as
+\subsection{Computation of $T$: FFT}. We compute T using a fast Fourier transform, defined as
 \begin{equation}
   \mathcal F(f)(n):=\sum_{m\in\mathcal N}e^{-\frac{2i\pi}{N_1}m_1n_1-\frac{2i\pi}{N_2}m_2n_2}f(m_1,m_2)
 \end{equation}
@@ -257,6 +311,11 @@ Therefore,
   \right)(k)
 \end{equation}
 
+\subsection{Observables}
+\indent
+We define the following observables.
+\bigskip
+
 \subsubsection{Energy}.
 We define the energy as
 \begin{equation}
@@ -367,7 +426,7 @@ The enstrophy is defined as
   .
 \end{equation}
 
-\subsubsection{Lyapunov exponents}
+\subsection{Lyapunov exponents}
 \indent
 To compute the Lyapunov exponents, we must first compute the Jacobian of $\hat u^{(n)}\mapsto\hat u^{(n+1)}$.
 This map is always of Runge-Kutta type, that is,
@@ -453,62 +512,6 @@ The Lyapunov exponents at time $\alpha\mathfrak L_r$ are then
   .
 \end{equation}
 
-\subsubsection{Numerical instability}.
-In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences.
-It is sufficient to ensure that the convolution term $T(\hat u,k)$ satisfies $T(\hat u,-k)=T(\hat u,k)^*$.
-After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results).
-
-\subsection{Reversible equation}
-\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}.
-It is defined directly in Fourier space:
-\begin{equation}
-  \partial_t\hat U_k=
-  -\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k
-  -i\frac{2\pi}L\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
-  (q\cdot\hat U_p)\hat U_q
-  ,\quad
-  k\cdot\hat U_k=0
-\end{equation}
-where $\alpha$ is chosen such that the enstrophy is constant.
-In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}):
-\begin{equation}
-  \partial_t\hat u_k=
-  -\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k
-  +\hat g_k
-  +\frac{4\pi^2}{L^2|k|}T(\hat u,k)
-  .
-  \label{rns_k}
-\end{equation}
-To compute $\alpha$, we use the constancy of the enstrophy:
-\begin{equation}
-  \sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k
-  =0
-\end{equation}
-which, in terms of $\hat u$ is
-\begin{equation}
-  \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\partial_t\hat u_k
-  =0
-\end{equation}
-that is
-\begin{equation}
-  \frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4|\hat u_k|^2
-  =
-  \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\hat g_k
-  +\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_k^*T(\hat u,k)
-\end{equation}
-and so
-\begin{equation}
-  \alpha(\hat u)
-  =\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
-  .
-  \label{alpha}
-\end{equation}
-Note that, by\-~(\ref{realu})-(\ref{realT}),
-\begin{equation}
-  \alpha(\hat u)\in\mathbb R
-  .
-\end{equation}
-
 
 
 \vfill