# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 4 Application to Incompressible Navier-Stokes Equations
### Spatial Discretization  
Equations (35) and (36) are firstly discretized on a cartesian spatial mesh, which will be here assumed to be uniform. As it is very common in the numerical discretization of incompressible Navier-Stokes equations, the variables are arranged on a staggered layout, in which velocity components are located on the normal faces of the cells and pressure nodes are positioned at the centers (Figure 5). The staggered arrangement of the variables has been often preferred to the simpler collocated or regular arrangements for incompressible flows, because it provides a more robust link between the discrete variables and avoids the odd-even decoupling of the pressure (also known as pressure checker-boarding problem) [26]. Moreover, it is well known that discrete operators based on explicit and symmetric central differences can easily provide primary (i.e., on mass and momentum) and secondary (e.g., on global kinetic energy) conservation on staggered grids [29]. This is an important topic, since a spatial discretization which is able to enforce conservation of global kinetic energy on a discrete level usually permits to avoid the onset of nonlinear instabilities arising from the accumulation of aliasing errors associated to the discrete evaluation of nonlinear terms [31; 32]. The well known second order Harlow-Welch discretization procedure is the simplest example of staggered discretization globally conserving kinetic energy. Moreover, it directly discretizes the _divergence_ form of the convective term, which has several advantages, in terms of local primary conservation properties and reduced computational cost. The Harlow-Welch procedure has been extended also to higher order central explicit schemes, both on uniform and variable meshes [29; 33]. When dealing with implicit (central) compact schemes,however, it is not clear if the discretization of the nonlinear convective term in Equation (35) can be performed directly on the divergence form and still retaining the global conservation of kinetic energy at the discrete level. In this case, the employment of the so-called _skew-symmetric_ splitting is mandatory, as in the case of regular layouts, since it automatically guarantees conservation of global kinetic energy on a semi-discrete level [29]. Recently, other approaches have been proposed, in which the beneficial properties of the skew-symmetric splitting are obtained by properly alternating the more economic divergence and advective forms inside the stages of a multistage time integration procedure [34; 35; 36]. In this paper, we will not consider these more sophisticated techniques, and will instead rely on the classical skew-symmetric form for the convective term. One-dimensional derivatives of zeroth, first, and second order are used to discretize spatial operators involved in the computation of the non-linear convective as well as linear viscous terms in Equation (35). While the latter does not require any special attention, the former does, due to its non linearity. On a discrete level, the skew-symmetric form leads a discretization of the \(i\)-component of the convective term in momentum equation Equation (35) of the form (cfr. Equations (87) and (89) of [PERSON] et al. [29]):  
\[\frac{1}{2}\left(\frac{\delta(\overline{u_{i}}^{x_{i}}\overline{u_{i}}^{x_{j} })}{\delta x_{j}}+\overline{\overline{u_{i}}^{x_{i}}\frac{\delta u_{i}}{ \delta x_{j}}}\right)\,, \tag{37}\]  
where \(\delta/\delta x_{j}\) is the finite difference derivative in direction \(x_{j}\) and overbars denote finite difference interpolations. In our case both differentiations and interpolations are performed by compact formulae. Hence, on a fully staggered arrangement of variables, the two addends in (37) are calculated by the following combination of interpolations and differentiantions:  
1. Divergence form 1. components \(i\) and \(j\) of velocity are interpolated in the direction \(j\) and \(i\) respectively; 2. the product is performed; 3. the staggered derivative of the product in direction \(j\) is computed. 2.