# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## Appendix A Modified Wavenumber  
The first derivative compact schemes from Lele [1] have the following formula:  
\[\beta f^{\prime}_{i-2}+\alpha f^{\prime}_{i-1}+f^{\prime}_{i}+\alpha f^{\prime}_ {i+1}+\beta f^{\prime}_{i+2}=a\frac{f_{i+1}-f_{i-1}}{2h}+b\frac{f_{i+2}-f_{i-2}} {4h}+c\frac{f_{i+3}-f_{i-3}}{6h}. \tag{11}\]  
From the Fourier analysis the modified wavenumber corresponding to Equation (11) is found to be  
\[w^{\prime}_{\mathrm{L}}(w)=\frac{a\sin(w)+\frac{b}{2}\sin(2w)+\frac{c}{3}\sin (3w)}{1+2\alpha\cos(w)+2\beta\cos(2w)}.\]  
First derivative Hermitian schemes allow the computation of the derivative at staggered points, given the values of the function and the values of its derivative at the meshpoints. They have the following general formula  
\[f^{\prime}_{i+1/2}=a\frac{f_{i+1}-f_{i}}{2h}+b\frac{f_{i+2}-f_{i-1}}{4h}+c \frac{f_{i+3}-f_{i-2}}{6h}+\beta f^{\prime}_{i-1}+\alpha f^{\prime}_{i}+\alpha f ^{\prime}_{i+1}+\beta f^{\prime}_{i+2}, \tag{12}\]  
which can be rewritten in symmetric form with the substitution \(i\mapsto i-\frac{1}{2}\),  
\[f^{\prime}_{i}=a\frac{f_{i+1/2}-f_{i-1/2}}{2h}+b\frac{f_{i+3/2}- f_{i-3/2}}{4h}+c\frac{f_{i+5/2}-f_{i-5/2}}{6h}+\\ +\beta f^{\prime}_{i-3/2}+\alpha f^{\prime}_{i-1/2}+\alpha f^{ \prime}_{i+1/2}+\beta f^{\prime}_{i+3/2}. \tag{13}\]  
The wavenumber corresponding to Equation (13) is easily found to be  
\[w^{\prime}_{\mathrm{H}}(w)=\frac{a}{h}\sin\left(\frac{w}{2}\right)+\frac{b}{2 h}\sin\left(\frac{3w}{2}\right)+\frac{c}{3h}\sin\left(\frac{5w}{2}\right)+\left[2 \alpha\cos\left(\frac{w}{2}\right)+2\beta\cos\left(\frac{3w}{2}\right)\right] w^{\prime}, \tag{14}\]  
where the derivatives at the RHS are supposed to be known exactly. Here we suppose, on the contrary, that only the values of the function (i.e., the values \(f_{i}\)) are assigned and that the compact scheme in Equation (11) is used to get a larger number of informations which are then used to feed the Hermitian interpolation subtending Equation (12). In this case, the wavenumber multiplying the square brackets in Equation (14) is \(w^{\prime}_{\mathrm{L}}\), the modified wavenumber pertaining the scheme used to obtain the values \(f^{\prime}_{i}\). \[w^{\prime}_{\mathrm{H}}(w)=\frac{a}{h}\sin\left(\frac{w}{2}\right)+\frac{b}{2 h}\sin\left(\frac{3w}{2}\right)+\frac{c}{3h}\sin\left(\frac{5w}{2}\right)+\left[2 \alpha\cos\left(\frac{w}{2}\right)+2\beta\cos\left(\frac{3w}{2}\right)\right] w^{\prime}_{\mathrm{L}} \tag{15}\]Similarly, interpolation Hermitian schemes allow the computation of the interpolant at staggered points, given the values of the function and the values of its derivative at the meshpoints, through the following formula  
\[f_{i+1/2}=a\frac{f_{i+1}+f_{i}}{2}+b\frac{f_{i+2}+f_{i-1}}{2}+c\frac{f_{i+3}+f_ {i-2}}{2}-\alpha\frac{f_{i+1}^{\prime}-f_{i}^{\prime}}{h}-\beta\frac{f_{i+2}^{ \prime}+f_{i-1}^{\prime}}{3h}, \tag{10}\]  
which can be again rewritten in symmetric form with the substitution \(i\mapsto i-\frac{1}{2}\),  
\[f_{i}=a\frac{f_{i+1/2}+f_{i-1/2}}{2}+b\frac{f_{i+3/2}+f_{i-3/2}}{ 2}+c\frac{f_{i+5/2}+f_{i-5/2}}{2}+\\ -\alpha\frac{f_{i+1/2}^{\prime}-f_{i-1/2}^{\prime}}{h}-\beta\frac {f_{i+3/2}^{\prime}-f_{i-3/2}^{\prime}}{3h}. \tag{11}\]  
The transfer function corresponding to Equation (11) is easily found to be  
\[T_{\rm H}(w)=a\cos\left(\frac{w}{2}\right)+b\cos\left(\frac{3w}{2}\right)+c \cos\left(\frac{5w}{2}\right)+2\left[\frac{\alpha}{h}\sin\left(\frac{w}{2} \right)+\frac{\beta}{3h}\sin\left(\frac{3w}{2}\right)\right]w^{\prime}, \tag{12}\]  
where the derivatives at the RHS are supposed to be known exactly. If we suppose, as in the previous case, that the compact scheme in Equation (10) is used to get a larger number of informations to feed the Hermitian interpolation subtending Equation (10), the wavenumber multiplying the square brackets in Equation (12) is \(w_{\rm L}^{\prime}\), the modified wavenumber pertaining the scheme used to obtain the values \(f_{i}^{\prime}\).