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
|
| 3/4 | -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 schemes are: Leapfrog, Euler and 2nd order Runge-Kutta scheme) is implemented as following (here for example the u-component of velocity):
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and steered by the array tsc(1:5)
| tsc(1) | tsc(2) | tsc(3) | tsc(4) | tsc(5)
|
| 1 | 1/3 | 0 | 0 | 0 | 1st substep
|
| 1 | 15/16 | -25/48 | 0 | 0 | 2nd substep
|
| 1 | 8/15 | 1/15 | 0 | 1 | 3rd 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 pressure solver and tsc(5) the 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.
