# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 3 Analysis of Novel Schemes
### Resolution Properties  
The resolution properties of the proposed schemes in the periodic case can be studied, as usual, by considering the scaled modified wavenumber \(w^{\prime}(w)\) associated to the numerical derivative. By assuming periodic boundary conditions, the dependent function \(f\) on the mesh can be Fourier-decomposed into its harmonic components \(\hat{f}\exp(iwx)\) where \(i=\sqrt{-1}\), \(w=2\pi kh/L\) and the wavenumber \(k\) assumes the integer values between \(-N/2\) and \(N/2\). The application of a finite-difference approximation of the first derivative to the generic Fourier component gives a coefficient \(iw^{\prime}\hat{f}\) in place of the exact derivative coefficient \(iw\hat{f}\), where \(w^{\prime}\) is real for central schemes. The discrepancy between \(w^{\prime}\) and \(w\) can be considered as a measure of the error in the numerical derivative. In the same way, the application of a second numerical derivative produces coefficients \(-w^{\prime\prime}\hat{f}\) which can be compared to exact second derivative coefficients \(-w^{2}\hat{f}\), while the application of an interpolation scheme to a single Fourier mode of frequency \(w\) introduces a transfer function \(T(w)\).  
In Figure 4a the scaled modified wavenumber of the proposed schemes for first derivative is compared to the ones relative to classical staggered fourth- and sixth-order tridiagonal compact schemes and to the collocated fourth-order Pade scheme. The modified wavenumber of the explicit fourth-order scheme given by Equation (3) is also shown for comparison. The legend for the curve labels is reported in Table 1. Figure 4a shows that, as compared to classical staggered compact schemes, obtained by maximizing the order of accuracy at fixed stencil, the proposed schemes have improved resolution properties, since the modified wavenumber plots stay close to the exact differentiation curve for a wider range of wavenumbers.  
A more quantitative comparison is provided in Table 2 for first derivative schemes, where the classical resolving efficiency \(e\) is reported for different values of the error tolerance \(\varepsilon\). The efficiency of the scheme is calculated as the shortest normalized wavenumber satisfying the relation  
\[\left|\frac{w^{\prime}-w}{w}\right|\leq\varepsilon. \tag{33}\]  
A further indication of the performances of the scheme is represented by the integral resolving efficiency \(e_{\mathrm{I}}\), here defined as  
\[e_{\mathrm{I}}=\int_{0}^{\pi}\left|w^{\prime}-w\right|^{k}\mathrm{d}w, \tag{34}\]  
that is often used with \(k=2\)[24; 25]. From Table 2 the good performances of the proposed schemes, in comparison with their classical counterparts, is confirmed. The performances of the new schemes are always superior to that of classical staggered schemes when they are measured with the integral resolving efficiency, while are comparable when measured by considering the shortest wavenumber satisfying Equation (33). Similar results, not shown here, were found for interpolation and second derivation formulae as well.  
Figure 4: Modified wavenumbers/transfer functions relative to several compact and Hermitian schemes, for interpolation, first derivative, and second derivative formulae. Refer to Table 1 for schemes legend. Solid and dashed lines are relative to novel and classical schemes respectively.  
The modified wavenumber of interpolation and second derivative schemes is presented in Figure 4b,c respectively. Again, the proposed schemes are compared to classical fourth- and sixth-order compact schemes and to fourth-order explicit schemes. The good performances with respect to classical compact schemes of the same order of accuracy is confirmed also in these cases.