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3rd order Runge-Kutta scheme

For the discretization in time a 3rd order low-storage Runge-Kutta scheme with 3 stages is used recommended by Williamson (1980). Generally an N-stage Runge-Kutta scheme discretizes an ordinary differential equation of the form

$\dfrac{d \psi}{d t} = f(t,\psi)$

as follows ( Baldauf, 2008 ):

\begin{align*} \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}. \end{align*}

The coefficients can be written in a so-called Butcher-Tableau:

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 applied 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

To save storage 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 (therefore low-storage scheme) as follows:

\begin{align*} \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) \end{align*}

For reasons of clarity the time integration for several schemes (further schemes are: Leapfrog, Euler and 2nd order Runge-Kutta scheme) is implemented as follows (here e.g. the u-component of velocity): 

u_p(k,j,i) = ( 1.0 - tsc(1) ) * u_m(k,j,i) + tsc(1) * u(k,j,i) + dt_3d * ( 
               tsc(2) * tend(k,j,i) + tsc(3) * tu_m(k,j,i)
             + tsc(4) * ( p(k,j,i) - p(k,j,i-1)) * ddx )
             - tsc(5) * rdf(k) * ( u(k,j,i) -ug )

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 previous 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. Aufl., 465 S.
  • Williamson, J. H., 1980: Low-storage Runge-Kutta schemes. J. Comput. Phys., 35, 48-56.
Last modified 3 years ago Last modified on Feb 23, 2021 3:49:18 PM
                                                                                                                                                                                                                                                                                                                                                                               
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