# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 2 Problem Formulation and Derivation of New Compact Schemes
Moreover, it will be useful to refer to a graphical representation of the various numerical schemes that will be introduced; each variable appearing in the finite-difference formula is depicted with a different symbol, depending on its order of derivation, and in a different position with respect to the stencil, depending on its role in the scheme. The explicit data \(f_{j}\) are depicted as small circles, while first and successive derivatives are denoted by a corresponding number of inclined dashes. If a variable is a known quantity in the formula, its position in the graphical representation falls within the lower part, below the horizontal line representing the mesh. If it is an unknown, it is positioned above the mesh, in the upper part of the sketch. In Figure 1, some examples are reported for different classical schemes, both explicit and compact. The weights appearing in the numerical formula can be also reported in the sketch as associated to each variable. When such weights are omitted, it is assumed that the drawing represents the (unique) maximum-order scheme attainable on that stencil. Figure 1a represents the classical second-order explicit scheme for the second derivative, while Figure 1b depicts the staggered explicit fourth-order scheme (4 SE-D1) given by:  
\[f_{j+1/2}^{\prime}=\frac{9}{8}\frac{f_{j+1}-f_{j}}{h}-\frac{1}{8}\frac{f_{j+2 }-f_{j-1}}{3h}. \tag{3}\]  
Also, Equation (1) is represented in Figure 1c, Equation (2) in Figure 1d. The procedure proposed in this paper is based on the observation that the Pade scheme here considered, Equation (1), can be obtained (as virtually all compact schemes) by evaluating the analytical derivative at the node \(x_{j}\) of a local polynomial reconstruction \(P_{j}\) for the unknown function \(f\), as it is done for explicit finite difference formulae. In the case of compact schemes, the implicit character of the relation between the set of grid nodes derivatives and the nodal values \(f_{j}\), suggests that local  
Figure 1: Graphical representation of explicit and compact schemes. **(a)** Classical second order explicit formula for second derivative. **(b)** Staggered explicit scheme of Equation (3). **(c)** Padé fourth order scheme, Equation (1). **(d)** Fourth order staggered compact derivative, Equation (2). polynomial representations of the unknown function \(f\) relative to adjacent nodes have to be linked by implicit relations. This representation is in general not unique. It is well known, for instance, that in the particular case of the Pade scheme, the classical third-degree cubic spline gives a consistent interpolation, in the sense that its analytical first derivative at node \(x_{j}\) returns Equation (1) [10]. In what follows, we will consider a different set of local interpolations which is consistent with the compact scheme in Equation (1). This set is given by the class of polynomials \(P_{j}\), of degree four, determined by imposing the explicit conditions  
\[P_{j}(x_{k})=f_{k}\qquad k=j-1,j,j+1, \tag{4}\]  
together with the implicit conditions  
\[P_{j}^{\prime}(x_{j+1}) = P_{j+1}^{\prime}(x_{j+1}), \tag{5}\] \[P_{j}^{\prime}(x_{j-1}) = P_{j-1}^{\prime}(x_{j-1}). \tag{6}\]  
It is easy to verify that the derivative at node \(x_{j}\) of the local polynomial expansion \(P_{j}\), which can be assumed to be defined on the interval \(\left[x_{j-1},x_{j+1}\right]\), furnishes again Equation (1). This approach was exploited in [16] in order to give an alternative derivation of many of the classical compact formulae. New interpolation procedures were also investigated in that paper, based on the enforcement of more general implicit conditions. The theory developed therein generalizes the classical spline interpolation, and led to new classes of compact-type schemes. In this paper, we adopt a slightly different standpoint, in which the implicit polynomial approach is not employed as a tool to justify the derivation of Equation (1), or of similar collocated formulae.