# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 2 Problem Formulation and Derivation of New Compact Schemes  
In this section, we present the procedure for obtaining the whole set of compact formulae needed in a typical staggered Navier-Stokes solver, starting from the computation of a single set of collocated compact derivatives. We will firstly describe how to construct the formuleby employing the classical fourth-order Pade scheme as a starting point. The extension to higher-order schemes is straightforward and will be presented in Section 2.4. Moreover, the procedure will be illustrated with reference to a uniform mesh; the extension to non-uniform meshes is also in principle straightforward, but will be not considered here. Hereinafter, when referring to derivatives, we will use the terms _collocated_ and _staggered_ to denote that the derivatives are computed in the same location, or shifted with respect to the original data set respectively. Let us assume that the dependent variable \(f\left(x\right)\) is discretized on an uniform mesh \(x_{j}=jh\) with \(j=0\ldots N\) and \(h=L/N\). For interior points, the classical fourth-order Pade scheme reads:  
\[\frac{1}{6}f_{j-1}^{\prime}+\frac{2}{3}f_{j}^{\prime}+\frac{1}{6}f_{j+1}^{ \prime}=\frac{f_{j+1}-f_{j-1}}{2h}, \tag{1}\]  
with truncation error \(-(1/180)h^{4}f^{V}\). This equation is a typical choice for high-order compact approximations of the first derivative on a collocated mesh. The set of nodal derivatives is obtained by solving a tridiagonal linear system, for which standard highly-efficient methods are available. The computational cost associated to this scheme, for a system of \(N\) equations, can be roughly estimated to be \(10N\) floating point operations. This value is obtained by summing the cost of the computation of the right-hand side (\(2N\)) to that of a standard Thomas solver for the tridiagonal system (\(8N\), [22]). Being a central method, the Pade scheme is free of numerical dissipation when employed in linear problems with periodic boundary conditions. In the non-periodic case it requires the specification of one non-symmetric scheme for each boundary node, in the case in which the dependent variable naturally falls on the boundary. In a typical staggered Navier-Stokes solver, however, this scheme cannot be used, since the quantities known at nodes \(x_{j}\) have to be either interpolated or differentiated at the staggered locations \(x_{j+1/2}=(x_{j}+x_{j+1})/2\). For this problem, the analogous staggered fourth order compact scheme is given by:  
\[\frac{1}{24}f_{j-1/2}^{\prime}+\frac{11}{12}f_{j+1/2}^{\prime}+\frac{1}{24}f_ {j+3/2}^{\prime}=\frac{f_{j+1}-f_{j}}{h}, \tag{2}\]  
with truncation error \(-(17/5760)h^{4}f^{V}\). Please note that in Equations (1) and (2), as well as in all the other classical compact formulae considered in the following, the convention that the left-hand side coefficients sum to one is assumed. This convention is alternative to the common choice for which the coefficient of the central term (\(f_{j}^{\prime}\) or \(f_{j+1/2}^{\prime}\)) equals to one; it has been adopted with the aim of obtaining a fair comparison of the truncation errors among the various compact and explicit formulae considered in this work. Equation (2) is again a centred non-dissipative scheme. Again, one non-symmetric scheme is required for each boundary node when the dependent variable falls on the boundary. The computational cost associated to this scheme can be again estimated to be \(10N\) operations. In what follows, each scheme will be referred to by a short acronym of the form \(n_{1}L_{1}L_{2}\)-D\(n_{2}\), where \(n_{1}\) is an integer equal to the formal order of accuracy of the scheme, \(L_{1}\) is either C or S for collocated and staggered schemes respectively, \(L_{2}\) is E, C, or H for explicit, compact, and proposed Hermitian schemes, and \(n_{2}\) is the order of derivation of the left-hand side of the scheme (0 for interpolation schemes). For instance, Equations (1) and (2) are referred to by the acronyms 4 CC-D1 and 4 SC-D1 respectively.