Changes between Version 5 and Version 6 of doc/tec/advection

Timestamp:

Jan 7, 2011 2:32:26 PM (15 years ago)

Author:

suehring

Comment:

--

doc/tec/advection

@@ -49 +49 @@
 }}}  
 the Courant number which characterizes stability properties and i is a complex number. 
-{{{
-#!Latex
-\[ \kappa_{eff} \]
-}}}
-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. \\\\
+κ,,eff,, 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 
-{{{
-#!Latex
-\[ \kappa\Delta x. \]
-}}}
-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 
-{{{
-#!Latex
-\[ \kappa\Delta x = 1.69 \]
-}}}
-in conjunction with the used Runge-Kutta method (Baldauf, 2008), the 5^th^ order dissipation is more than sufficient to avoid instabilities.
+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 κ Δx.
+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-Δx. 
+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-Δx with WS5, whereat with WS3 16-Δx waves are still affected. Accordingly to a maximum of the amplification factor at κ Δx = 1.69 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).