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Timestamp:: May 10, 2016 5:23:52 PM (9 years ago)
Author:: Giersch
Comment:: --

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doc/tec/discret

-                      v10
+                      v11
 It is thus possible to calculate the derivatives of the velocity components at the center of the volumes (same location as the scalars). By the same token, derivatives of scalar quantities can be calculated at the edges of the volumes. In this way it is possible to calculate derivatives over only one grid length and the effective spatial model resolution can be increased by a factor of two in comparison to non-staggered grids.
 By default, the advection terms in the first five equations in Sect. [wiki:doc/tec/gov governing equations] are discretized using an upwind-biased 5th-order differencing scheme in combination with a 3rd-order Runge–Kutta  time-stepping scheme after [#williamson Williamson (1980)]. [#wicker Wicker and Skamarock(2002)] compared different time- and advection differencing schemes and found that this combination give the best results regarding accuracy and algorithmic simplicity. However, the 5th-order differencing scheme is known to be overly dissipative. It is thus also possible to use a 2nd-order scheme after [#piacsek Piacsek and Williams(1970)]. The latter scheme is non-dissipative, but it suffers from immense numerical dispersion. Time discretization can also be achieved using 2nd-order Runge–Kutta or 1st-order Euler schemes.
+By default, the advection terms in the first five equations in Sect. [wiki:doc/tec/gov governing equations] are discretized using an upwind-biased 5th-order differencing scheme in combination with a 3rd-order Runge–Kutta  time-stepping scheme after [#williamson Williamson (1980)]. [#wicker Wicker and Skamarock(2002)] compared different time- and advection differencing schemes and found that this combination give the best results regarding accuracy and algorithmic simplicity. However, the 5th-order differencing scheme is known to be overly dissipative. It is thus also possible to use a 2nd-order scheme after [#piacsek Piacsek and Williams(1970)]. The latter scheme is non-dissipative, but it suffers from immense numerical dispersion. Time discretization can also be achieved using 2nd-order Runge–Kutta or 1st-order Euler schemes. For more information about the advection and time differencing schemes see [wiki:doc/tec/discret#Higherorderadvectionscheme Higher order advection scheme] and [wiki:doc/tec/discret#Timeintegration time integration].
 == Higher order advection scheme ==
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 Figure 2: The dispersion and dissipation error as a function of the dimensionless wavenumber ''κ Δx'' for WS3 (3^rd^ order scheme), WS4 (4^th^ order scheme), WS5, WS6 and the 2^nd^ order scheme of [#piacsek Piacsek and Williams (1970)] (PW).
 Fig. 2 shows that the dispersion error of the upwind schemes and the dispersion error of the next higher, even ordered scheme are identical. Generally the dispersion error decreases with increasing order of the discretization. However, no of the depicted schemes is able to adequately resolve structures with wavelengths near 2-''Δx'' (generally no scheme based on finite differences is capable to do this). The centered, even ordered schemes hold no dissipation errors. With increasing order the numerical dissipation is more local. So the maximum wavelength affected by the dissipation term is round about 8-''Δx'' with WS5, whereas wavelength of 16-''Δx'' are still affected with WS3. Accordingly to the maximum of the amplification factor at ''κ Δx'' = 1.69 (these waves become unstable at first) in conjunction with the used [wiki:doc/tec/discret#3^rd^orderRunge-Kuttascheme Runge-Kutta method] ([#baldauf Baldauf, 2008]), the 5^th^ order dissipation is sufficient to avoid instabilities. The maximum stable Courant-number is C,,r,, = 1.4 ([#baldauf Baldauf, 2008]).
+Fig. 2 shows that the dispersion error of the upwind schemes and the dispersion error of the next higher, even ordered scheme are identical. Generally the dispersion error decreases with increasing order of the discretization. However, no of the depicted schemes is able to adequately resolve structures with wavelengths near 2-''Δx'' (generally no scheme based on finite differences is capable to do this). The centered, even ordered schemes hold no dissipation errors. With increasing order the numerical dissipation is more local. So the maximum wavelength affected by the dissipation term is round about 8-''Δx'' with WS5, whereas wavelength of 16-''Δx'' are still affected with WS3. Accordingly to the maximum of the amplification factor at ''κ Δx'' = 1.69 (these waves become unstable at first) in conjunction with the used [wiki:doc/tec/discret#a3rdorderRunge-Kuttascheme Runge-Kutta method] ([#baldauf Baldauf, 2008]), the 5^th^ order dissipation is sufficient to avoid instabilities. The maximum stable Courant-number is C,,r,, = 1.4 ([#baldauf Baldauf, 2008]).
 '''Note: A stable numerical solution can only be guaranteed with the 3 rd order [../rk3 Runge-Kutta method].'''
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 where ''u'' denotes the corresponding velocity component. Furthermore, the turbulent fluxes are evaluated on each Runge-Kutta substep and weighted with the respective Runge-Kutta coefficients to remove dependencies of the Runge-Kutta substeps. The interpretation of the turbulent fluxes as variances and covariances is no longer valid when using WS5. For other advection schemes, like the PW-scheme, the interpretation of turbulent fluxes as co/variances is still valid, because the discretization is alike the computation of the co/variances.
 = 3^rd^ order Runge-Kutta scheme =
+= Time integration =
 For the discretization in time a 3^rd^ order low-storage Runge-Kutta scheme with 3 stages is used recommended by [#williamson Williamson (1980)]. Generally an N-stage Runge-Kutta scheme discretizes an ordinary differential equation of the form