3rd 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
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as follows ( Baldauf, 2008 ):
![\[ \psi^{(0)} = \psi^{n}, \]
\[ k^{i} = f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \]
\[ \psi^{i} = \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \]
\[ \psi^{n+1} = \psi^{N}. \]](/trac/tracmath/22007c852ab1a11e37ad031ac90c189230b8e234.png)
The coefficients can written in a Butcher-Tableau in the following way:
| α1 | β1,1 | 0 | ... | |
|
| α2 | β2,1 | β2,2 | 0 | ... |
|
| ... | ... | | | |
|
| αN | βN,1 | βN,2 | ... | βN,N-1 | 0
|
| | βN+1,1 | βN+1,2 | ... | βN+1,N-1 | βN+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):
