Details on RK in doc

docs/nstrophy_doc.tex

@@ -182,6 +182,18 @@ Note that, by\-~(\ref{realu})-(\ref{realT}),
 
 \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 $\hat u_k(t_n)$.
+These algorithms approximate the solution to an equation of the form $\dot u=f(t;u)$ with
+\begin{equation}
+  \hat u_k^{(n+1)}=\hat u_k^{(n)}
+  +\delta_n\sum_{i=1}^sb_ik_i(t_n;\hat u^{(n)})
+  ,\quad
+  k_i(t_n;\hat u^{(n)}):=f\left(t_n+c_i\delta_n;\ \hat u+\delta_n\sum_{j=1}^{i-1}a_{i,j}k_j(t_n,\hat u^{(n)})\right)
+  .
+\end{equation}
+The $c_i$ and $a_{i,j}$ are chosen in one of various ways, depending on the desired accuracy.
+\bigskip
+
+\indent
 {\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}
@@ -428,6 +440,11 @@ The enstrophy is defined as
 
 \subsection{Lyapunov exponents}
 \indent
+We will consider several methods of computing the Lyapunov exponents.
+\bigskip
+
+\subsubsection{Method 1: compute the Jacobian at every step}
+\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,
 \begin{equation}