Discretisation¶
Normalisation¶
As derived in the governing equation, there are \(2 N\) unknowns
\[ \begin{align}\begin{aligned}& \vel_{0}, \vel_{1}, \cdots, \vel_{N - 2}, \vel_{N - 1},\\& \pos_{0}, \pos_{1}, \cdots, \pos_{N - 2}, \pos_{N - 1},\end{aligned}\end{align} \]
and \(2 N\) equations:
\[ \begin{align}\begin{aligned}&
\tder{\pos_{\ia}}{t}
=
\vel_{\ia},\\&
\lag
=
0_{\ia}.\end{aligned}\end{align} \]
In addition to the fact that the equations being \(m\)-independent, I normalise these equations by the reference length scale \(l\) and by the reference time scale \(\sqrt{l / g}\), giving
\[ \begin{align}\begin{aligned}&
\tder{\pos_{\ia}}{t}
=
\vel_{\ia},\\&
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\tder{\vel_{\ib}}{t}
\cos \left( \pos_{\ia} - \pos_{\ib} \right)
+
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\vel_{\ib}
\vel_{\ib}
\sin \left( \pos_{\ia} - \pos_{\ib} \right)
-
\left( N - \ia \right)
\cos \pos_{\ia}
=
0_{\ia},\end{aligned}\end{align} \]
which are treated in the following part.
Scheme¶
The simplest scheme would be:
\[ \begin{align}\begin{aligned}&
\dder{\pos_{\ia}}{t}
=
\vel_{\ia}^{n+1},\\&
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\dder{\vel_{\ib}}{t}
\cos \left( \pos_{\ia}^n - \pos_{\ib}^n \right)
=
\left( N - \ia \right)
\cos \pos_{\ia}^n
-
\sum_{\ib = 0}^{N - 1}
\left\{ N - \max \left( \ia, \ib \right) \right\}
\vel_{\ib}^n
\vel_{\ib}^n
\sin \left( \pos_{\ia}^n - \pos_{\ib}^n \right),\end{aligned}\end{align} \]
which requests to solve only one linear system.
This scheme, however, does not satisfy the energy conservation in general (see Example).
The first part is devoted to find a proper scheme which is discretely energy-conserving, which is followed by the concrete numerical treatment.