| 77 | | Fig. 5 shows 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). 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 [../rk3 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]). |
| | 77 | 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). |
| | 78 | |
| | 79 | 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 [../rk3 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]). |
