# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 3 Analysis of Novel Schemes  
### Structure of the Schemes  
The structure of the novel global schemes can be more conveniently illustrated by resorting to matrix notation. With reference to the staggered derivative in the fourth-order case, the compact scheme (1), together with appropriate boundary conditions, can be expressed as:  
\[\mathbf{A}\mathbf{f}^{\prime}=\mathbf{B}\mathbf{f}, \tag{26}\]  
where \(\mathbf{A}\) and \(\mathbf{B}\) have tridiagonal structure at interior points. The formula for the staggered derivative based on Hermitian interpolation, Equation (7), can be expressed as:  
\[\mathbf{f}_{S}^{\prime}=\mathbf{H}\mathbf{f}+\mathbf{K}\mathbf{f}^{\prime}, \tag{27}\]  
where \(\mathbf{f}_{S}^{\prime}\) is the set of staggered derivatives, and the matrices \(\mathbf{H}\) and \(\mathbf{K}\) have two non-zero diagonals in the fourth-order case, and are typically rectangular for non-periodic problems. The global scheme is then formally expressed by  
\[\mathbf{f}_{S}^{\prime}=\left(\mathbf{H}+\mathbf{K}\mathbf{A}^{-1}\mathbf{B} \right)\mathbf{f}. \tag{28}\]  
Similar formulae can be written for the cases of interpolation and collocated second derivative and for higher-order schemes.  
In the particular case in which the problem is discretized on a uniform mesh with periodic boundary conditions, all the matrices in Equation (28) are square and circulant, and, as such, they commute. In light of this, the multiplication of Equation (28) by \(\mathbf{A}\) to the left yields  
\[\mathbf{A}\mathbf{f}_{S}^{\prime}=\left(\mathbf{A}\mathbf{H}+\mathbf{K} \mathbf{B}\right)\mathbf{f}. \tag{29}\]  
where the matrix \(\mathbf{A}\mathbf{H}+\mathbf{K}\mathbf{B}\) has four non-zero diagonals in the fourth-order case. This analysis indicates that for periodic problems the scheme for the staggered derivative given by the application of Equations (1) and (7) is equivalent to the fourth-order staggered compact scheme:  
Figure 3: Graphical representation of new schemes. (**a**) Staggered sixth order Hermitian derivative scheme (Equation (15)). (**b**) Staggered sixth order interpolation scheme (Equation (8)). (**c**) Collocated sixth order second derivative scheme (Equation (22)).  
\[\frac{1}{6}f^{\prime}_{j-1/2}+\frac{2}{3}f^{\prime}_{j+1/2}+\frac{1}{6}f^{\prime}_ {j+3/2}=\frac{5}{8}\frac{f_{j+1}-f_{j}}{h}+\frac{3}{8}\frac{f_{j+2}-f_{j-1}}{3h}. \tag{30}\]  
A similar analysis shows that in the case of interpolation the application of Equations (1) and (9) is equivalent to the scheme:  
\[\frac{1}{6}f_{j-1/2}+\frac{2}{3}f_{j+1/2}+\frac{1}{6}f_{j+3/2}=\frac{23}{48} \left(f_{j}+f_{j+1}\right)+\frac{1}{48}\left(f_{j-1}+f_{j+2}\right), \tag{31}\]  
while the application of Equations (1) and (11) is equivalent to the scheme:  
\[\frac{1}{6}f^{\prime\prime}_{j-1}+\frac{2}{3}f^{\prime\prime}_{j}+\frac{1}{6} f^{\prime\prime}_{j+1}=\frac{2}{3}\frac{f_{j-1}-2f_{j}+f_{j+1}}{h^{2}}+\frac{1}{3} \frac{f_{j-2}-2f_{j}+f_{j+2}}{4h^{2}}. \tag{32}\]  
All these schemes are members of the fourth-order families of formulae for staggered first derivatives and interpolations and for collocated second derivatives. Each scheme is a suboptimal element of this family, since its coefficients are not optimized with the aim of obtaining the maximum order attainable by the family. In this sense, the procedure developed in this paper can be seen, in the periodic case, as a prefactorization, by means of which particular suboptimal schemes can be selected.  
It is worth noting that this equivalence is valid only in the periodic case. In non-periodic applications, the classical formulae require individual closures near boundaries, while the proposed method needs only single closure schemes for the collocated first derivative, while the Hermitian formulae based on values and first derivatives, Equations (7), (9) and (11) for the fourth-order case, remain unaltered. In this last case the new method produces completely new schemes, whose quality will be discussed in Section 5.