# Construction of Compact Finite Difference Schemes by Classic Differential Quadrature
## 4 Fourier Stability Analysis for the Derived Compact Difference Schemes  
Stability analysis is an important part of the numerical integration approach. In general, we use linear stability analysis by the use of a standard scalar test equation (\(\dot{x}=\lambda x\)) [21; 22; 23]. However, the linear stability analysis does not provide information about the performance of schemes at different length scales. Unlike this, the Fourier stability analysis provides an effective way to quantify the resolution characteristics such as dispersion or phase error of the differencing approximations. In this section, we analyze the dispersion and dissipation characteristics of the derived difference schemes using Fourier stability analysis [1; 24]. By the Fourier stability analysis, different numerical schemes have different estimates of the modified scaled wavenumber, and it is in general a complex quantity [25]. It can easily be shown that the real part of the modified scaled wavenumber tends to amplify/attenuate the corresponding Fourier mode while the imaginary part introduces a phase error. Thus the deviations of the real and imaginary parts of the modified scaled wavenumber from the exact scaled wavenumber can be respectively taken to be the indicators of numerical dissipation and dispersion. Furthermore, for numerical stability of any scheme, one must look at the imaginary part of the modified scaled wavenumber. The imaginary part of the modified scaled wavenumber represents numerical dissipation only when it is negative. Any scheme that produces a positive imaginary part of the modified scaled wavenumber is numerically unstable because a positive imaginary part is equivalent to adding anti-diffusion [25]. Figure 3 shows the modified scaled wavenumber of GBDF based on uniform grids. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(s\) & \(v\) & \(\omega_{0}\) & \(\omega_{1}\) & \(\omega_{2}\) & \(\omega_{3}\) & \(\omega_{4}\) \\ \hline
1 & 1 & \(-1\) & & & & \\
3 & 2 & \(-\frac{1}{4}\) & \(-\frac{9}{2}\) & & & \\
5 & 3 & \(\frac{1}{24}\) & \(-\frac{5}{5}\) & \(-\frac{10}{10}\) & & \\
7 & 4 & \(-\frac{1}{120}\) & \(\frac{20}{10}\) & \(-\frac{21}{20}\) & \(-\frac{35}{8}\) & \\
9 & 5 & \(\frac{500}{500}\) & \(-\frac{1}{112}\) & \(\frac{20}{14}\) & \(-\frac{2}{20}\) & \(-\frac{27}{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Coefficients of extended trapezoidal rules of the second kind (ETR\({}_{2}\)) using uniform grids. Based on the above results, we can see that GBDF are all stable numerical methods, and this kind of method has a strong numerical dissipation and dispersion. Figure 4 shows the modified scaled wavenumber of central compact difference schemes (CCDS) based on uniform grids. Based on the above results, we can see that CCDS have a strong numerical dispersion but have almost no numerical dissipation. Figure 5a,b show the modified scaled wavenumber of GAMs based on uniform grids and respectively for even \(s\) and odd \(s\). Figure 4: Plot of real and imaginary parts of the modified wavenumber for central compact difference schemes (CCDS). Figure 5: \(Cont.\)  
Figure 3: Plot of real and imaginary parts of the modified wavenumber for GBDF. Figure 6 shows the modified scaled wavenumber of \(\text{ETR}_{1}\) based on uniform grids. Based on the above results, we can see that the \(\text{ETR}_{1}\) are roughly equivalent to the GAMs with odd \(s\) in terms of numerical dissipation and dispersion. In fact, as mentioned earlier, these two methods are essentially the same. Figure 7 shows the modified scaled wavenumber of \(\text{ETR}_{2}\) based on uniform grids. Figure 5: (a) Plot of real and imaginary parts of the modified wavenumber for GAMs with even \(s\); (b) plot of real and imaginary parts of the modified wavenumber for GAMs with odd \(s\). Figure 6: Plot of real and imaginary parts of the modified wavenumber for \(\text{ETR}_{1}\). Figure 7: Plot of real and imaginary parts of the modified wavenumber for \(\text{ETR}_{2}\).