= 3^rd^ order Runge-Kutta scheme =

For the dicretization in time a 3 rd order low storage Runge-Kutta scheme with 3 stages recommended by Williamson ( 1979 ) is used.
Generally an N-stage Runge-Kutta scheme discretizes an ordinary differential equation of the form
{{{
#!Latex
\[  \frac{d \psi}{d t} & = & f(t,\psi) \]
}}}
as follows ( Baldauf, 2008 ):
{{{
#!Latex
\[ \psi^{(0)} = \psi^{n}, \]
\[ k^{i} = f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \]
\[ \psi^{i} = \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \]
\[ \psi^{n+1} = \psi^{N}. \]
}}}
The coefficients can written in a Butcher-Tableau in the following way:
|| α,,1,, || β,,1,1,, || 0 || ... || || ||
|| α,,2,, || β,,2,1,, || β,,2,2,, || 0 || ... || ||
|| ...    ||  ... || ||  ||  || ||
|| α,,N,, || β,,N,1,, || β,,N,2,, || ... || β,,N,N-1,, || 0 || 
||        || β,,N+1,1,, || β,,N+1,2,, || ... || β,,N+1,N-1,, ||  β,,N+1,N,, ||

The appendant coefficients for the used Runge-Kutta scheme reads:
||   0  ||   0   ||   0   ||   0  ||
||  1/3 ||    1/3 ||    0  ||  0   ||
|| 1/2  ||  -3/16 || 15/16 ||  0  ||
||      ||  1/6  ||  3/10 || 8/15 ||
…
For the implementation it is advantageous to compute PSI_N from the intermediate solutions PSI_1 and PSI_2 as follows:
… 
Due to the fact that the current tendency F(psi) is only required locally in time, the arrays F(Psi_n) and F(Psi_1) can combined after the 2 nd substep and buffered in F(psi_1):