= Higher order advection scheme =

Based on a flux formulation of the advection term 
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#!Latex
\[ \frac{\partial \psi}{\partial t} & = & -\frac{\partial ( u \psi )}{\partial x}, \] 
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the one dimensional advection equation can written in following semidiscrete form: 
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\[ \frac{\partial  \psi_{i} }{\partial t} = - \frac{F_{i+\frac{1}{2}}(u\psi) - F_{i-\frac{1}{2}}(u\psi)}{\Delta x}, \]
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where 
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\[F_{i\pm\frac{1}{2}} \]
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denotes the fluxes staggered a half grid length related to the advected quantity. \\
Wicker and Skamarock (2002) dicretized the 6^th^ and 5^th^ order fluxes as follows:
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\[ F_{i-\frac{1}{2}}^{6} &=& \frac{u_{i-\frac{1}{2}}}{60} \left[ 37(\psi_{i}+\psi_{i-1}) - 8(\psi_{i+1} + \psi_{i-2}) +(\psi_{i+2} + \psi_{i-3}) \right] \] 
}}}
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#!Latex
\[ F_{i-\frac{1}{2}}^{5} &=& F_{i-\frac{1}{2}}^{6} - \frac{|u_{i-\frac{1}{2}}|}{60} \left[10(\psi_{i}-\psi_{i-1}) -5(\psi_{i+1} - \psi_{i-2})-(\psi_{i+2} - \psi_{i-3}) \right] \]
}}}
The 5^th^ order upwind discretization (WS5) consists of the centered non dissipative 6^th^ (WS6) order flux and an artificial added numerical dissipation term. This term is necessary to stabilize the numerical solution, because higher order centered fluxes exhibits worse stability properties.The absolute value assures a dissipative effect also for u < 0. 

=== Numerical properties ===

A semidiscrete fourier transformation for the spatial derivatives maps the one dimensional advection equation in fourier space as follows (Baldauf, 2008): 
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\[ \frac{\partial \hat{\psi}_{\kappa}}{\partial t}  =  - \frac{i}{\Delta t} C_{r} \kappa_{eff}\,\hat{\psi}_{\kappa}, \]
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where
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#!Latex
\[
\hat{\psi}  
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denotes the fourier transformed of
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#!Latex
\[ 
\psi. \quad C_{r} = \frac{u \Delta t}{\Delta x} 
\]
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the Courant number which characterizes stability properties and i is a complex number. 
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\[ \kappa_{eff} \]
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is the effective wavenumber of a mode in fourier space which characterizes the modified wavenumber through the discretization. The real part of the effective wavenumber describes the dispersion error, the imaginary part the dissipation error. \\\\
[[Image(prop.png, 500px, border=1)]] \\
Fig. 1 shows the dispersion and dissipation errors of WS3, WS4, WS5, WS6 and the 2^nd^ order scheme of Piascek and Williams (1970) (PW) presented as a function of the dimensionless wavenumber 
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\[ \kappa\Delta x. \]
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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 dicretization. However, no scheme is able to resolute wavelength with the range near 2-Delta x. 
The centered even ordered schemes holds no dissipation errors. The numerical dissipation is more local with increasing order, so the maximal affected wavelength by the dissipation term is round about 8 gridlength with WS5, whereat with WS3 16 gridlength waves are still affected. Accordingly to a maximum of the amplification factor at 
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\[ \kappa\Delta x = 1.69 \]
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in conjunction with the used Runge-Kutta method (Baldauf, 2008), the 5^th^ order dissipation is more than sufficient to avoid instabilities.
The maximal stable Courant-number is C,,r,, = 1.4 (Wicker and Skamarock, 2002), (Baldauf, 2008).

'''Note: A stable numerical solution can only guaranteed with the 3 rd oder Runge-Kutta method.'''

=== Boundaries ===

Due to the large stencil of WS5 additional ghost layers are necessary on each lateral boundary of each processor subdomain to avoid local data dependencies. Therefor the exchange of ghost layers is adapted to a dynamic number of ghost layers. 
For the bottom and top boundaries a successive degradation from WS5 to WS3 to a 2^nd^ order scheme is required to avoid unphysical fluxes which would arise from the bottom and top ghost layers. The used 2^nd^ order scheme is based on a flux discretization to ensure consistency with the WS-schemes. So the skew symmetric PW-scheme cannot be used. Furthermore an additional numerical dissipation term of 2^nd^ order, based on (Shchepetkin and McWilliams, 1998),is required for the 2^nd^ order scheme to perform a numerical stable switching of advection schemes of different order.
These successive degradation is also done for the lateral radiation boundary condition at the outflow and near topography (Note: Topography is not implemented at the moment.) 


=== Statical evaluation of turbulent fluxes ===

The evaluation of turbulent fluxes should be consistent to the discretization in the prognostic equations, else some unphysical effects will occur. For example, the computation of the fluxes as variances and covariances will induce some conspicuous kinks in vertical heat and momentum fluxes near the surface, while the temperature and velocity profiles show no conspicuity. For computing turbulent fluxes as appearing in the prognostic equations the computed fluxes in the advection routines are buffered and reused also for the statistics. For getting only the turbulent, not the mean signal and to remove the influence of Galilei transformation, the centered flux F,,i+1/2,, has to be multiplied with a factor 
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\[ \frac{u_{i+\frac{1}{2}} - 2 \overline u}{u_{i+\frac{1}{2}} - u_{i, Galilei}} \] 
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and the dissipative flux with a factor
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#!Latex
\[ \frac{|u_{i+\frac{1}{2}} - 2 \overline u|}{|u_{i+\frac{1}{2}} - u_{i, Galilei}|}, \] 
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where u denotes the respective 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 substep. 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.