3rd order Runge-Kutta scheme
For the dicretization in time a 3rd 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 ψN from the intermediate solutions ψ1 and ψ2 and combine the local tendencies in one array after the second substep to save storage (therefor low-storage scheme) as follows:
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For reasons of clarity the time integration for several schemes ( further scheme are: Leap-frog, Euler and 2nd order Runge-Kutta scheme ) is implemented as following ( here for example for the u-component of velocity ):
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and steered by the array tsc
Sorry for incompleteness. A further description will follow in the next days.
