= 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}}}   ||
|| {{{3/4}}}  ||  {{{-3/16}}} || {{{15/16}}} ||  {{{0}}}  ||
||      ||  {{{1/6}}}  ||  {{{3/10}}} || {{{8/15}}} ||

For the implementation it is advantageous to compute ψ^N^ from the intermediate solutions ψ^1^ and ψ^2^ and combine the local tendencies in one array after the second substep to save storage (therefor low-storage scheme) as follows: 
{{{
#!Latex
\[ \hat\psi_{1} & = & \psi_{n} + \frac{1}{3} \Delta t f\left(\psi_{n}\right) \]
\[ \hat\psi_{2} & = & \hat\psi_{1} + \frac{1}{48} \Delta t \left( 45 f\left(\hat\psi_1\right) - 25 f\left(\psi_{n}\right) \right) \]
\[ f\left(\hat\psi_{1}\right) & = &-153 f\left(\hat\psi_{1}\right) + 85 f\left(\psi_{n}\right) \]
\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 15 f \left(\hat\psi_{1}\right) \right) \]
}}}

For reasons of clarity the [../../app/inipar#timestep_scheme time integration] for several schemes (further schemes are: Leapfrog, Euler and 2^nd^ order Runge-Kutta scheme) is implemented as following (here for example the u-component of velocity):

{{{
#!Latex
\begin{split}
 \textnormal{u}\_\textnormal{p}\left(k,j,i\right) = \left(1.0 - \textnormal{tsc}\left(1\right) \right) * \textnormal{u}\_\textnormal{m}\left(k,j,i\right) + \textnormal{tsc}\left(1\right) * \textnormal{u}\left(k,j,i\right) + \textnormal{dt}\_\textnormal{3d}* \left( \\ 
                             \textnormal{tsc}\left(2\right) * \textnormal{tend}\left(k,j,i\right) + \textnormal{tsc}\left(3\right) * \textnormal{tu}\_\textnormal{m}\left(k,j,i\right)  \\
                             + \textnormal{tsc}\left(4\right) * \left(\textnormal{p}\left(k,j,i)\right) - \textnormal{p}\left(k,j,i-1\right) \right) * \textnormal{ddx} \ \ ) \\
                             - \textnormal{tsc}\left(5\right) * \textnormal{rdf}\left(k\right) * \left(\textnormal{u}\left(k,j,i) - \textnormal{ug} \right) 
\end{split}
}}}

and steered by the array {{{tsc(1:5)}}}

|| {{{tsc(1)}}} || {{{tsc(2)}}} || {{{tsc(3)}}} || {{{tsc(4)}}} || {{{tsc(5)}}} ||
|| {{{1}}} || {{{1/3}}} || {{{0}}} || {{{0}}} || {{{0}}} || 1^st^ substep
|| {{{1}}} || {{{15/16}}} || {{{-25/48}}} || {{{0}}} || {{{0}}} || 2^nd^ substep
|| {{{1}}} || {{{8/15}}} || {{{1/15}}} || {{{0}}} || {{{1}}} || 3^rd^ substep

{{{u_p}}} is the prognosticated and {{{u}}} the current velocity at each substep. {{{u_m}}} denotes the velocity of the last substep (needed for Leapfrog). {{{tend}}} is the current tendency and {{{tu_m}}} the combined tendencies of the prior substeps. {{{tsc(4)}}} steers the preconditioning of the [../../app/inipar#psolver pressure solver] and {{{tsc(5)}}} the [../../app/inipar#rayleigh_damping_factor rayleigh damping].  


=== References ===  


* '''Baldauf, M., 2008:''' Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration. ''J. Comput. Phys.'', '''227''', 6638-6659.

* '''Durran, D. R., 1999:''' ''Numerical methods for wave equations in geophysical fluid dynamics.'' Springer Verlag, New York, 1. Auf\/l., 465 S.

* '''Williamson, J. H., 1980:''' Low-storage Runge-Kutta schemes. ''J. Comput. Phys.'', '''35''', 48-56.