# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 2 Problem Formulation and Derivation of New Compact Schemes
### Schemes for Staggered First Derivative  
The numerical formulae for the first derivatives \(f_{j+1/2}^{\prime}\), at staggered grid points \(x_{j+1/2}\), based on the Hermite polynomial \(P_{j}\), can be directly evaluated as a linear combination of nodal values \(f_{k}\) and nodal first derivatives \(f_{k}^{\prime}\) for \(k=j-1,j,j+1\). Since the six values \(f_{j-1},f_{j},f_{j+1}\) and \(f_{j-1}^{\prime},f_{j}^{\prime},f_{j+1}^{\prime}\), are related by Equation (1), only five of them can be chosen independently for the construction of the the fourth-degree polynomial \(P_{j}\). Although any choice of five data among them is equivalent, for what concerns the evaluation of first derivative at the staggered point \(x_{i+1/2}\), the selection of the values\(f_{j-1},f_{j},f_{j+1}\) and \(f^{\prime}_{j},f^{\prime}_{j+1}\) is particularly convenient, since for this set of data the coefficient of the value \(f_{j-1}\) results to be zero. This is a consequence of the symmetry of the problem: the first derivative at \(x_{j+1/2}\) of the third-degree polynomial fitting the data \(f_{j},f_{j+1}\) and \(f^{\prime}_{j},f^{\prime}_{j+1}\) is coincident with the derivative at the same point of the whole family of fourth-degree polynomials fitting the same data. This is similar to what happens for a linear interpolation between two points \(x_{j}\) and \(x_{j+1}\), which shares the same first derivative at \(x_{j+1/2}\) with the whole family of parabolas that fit the same data.  
For uniform meshes, the formula to compute the staggered first derivative based on collocated data and first derivatives (4 SH-D1, H standing for _Hermitian_) assumes the simple form:  
\[f^{\prime}_{j+1/2}=3\frac{f_{j+1}-f_{j}}{2h}-\frac{f^{\prime}_{j}+f^{\prime}_{ j+1}}{4}. \tag{7}\]  
The combination of Equations (1) and (7) gives a fourth-order scheme for the staggered first derivative at interior points with global truncation error \(-(13/5760)h^{4}f^{V}\). The standard reference for comparison for this scheme is the staggered fourth order Pade scheme of Equation (2), which has a comparable truncation error, while the explicit scheme of Equation (3) has truncation error \((9/1920)h^{4}f^{V}\). For non-periodic boundary conditions, Equation (1) has to be replaced by suitable non-symmetric compact formulae at the boundary nodes, while Equation (7) remains unaltered near boundaries. The computational cost of Equation (7) can be estimated to be \(5N\). The graphical sketch of the proposed scheme, which is built upon the successive application of Equations (1) and (7), is given in Figure 2a.  
A quick inspection of Equations (1) and (7) reveals that both express essentially the same relation between the five values in the sets \((f^{\prime}_{j-1},f^{\prime}_{j},f^{\prime}_{j+1},f_{j-1},f_{j+1})\) and \((f^{\prime}_{j},f^{\prime}_{j+1/2},f^{\prime}_{j+1},f_{j},f_{j+1})\) respectively, the only difference being the spacing (\(h\) in the former, \(h/2\) in the latter). Nonetheless, the two schemes differ in the crucial choice of known and unknown terms in the aforementioned sets, thus the two corresponding equations are presented separately ([19], cf. Equations (20) and (21)). We take the opportunity to stress that any finite difference formula can be obtained by requesting that a linear combination of the desired nodal values--of derivatives of various orders--be proportional to \(\mathcal{O}(h^{p})\), being \(p\) the minimum order of accuracy of the scheme. This approach is the most general and easily allows the computation of the coefficients used in several class of unconventional schemes [14; 15; 23].