# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 6 Conclusions  
In the context of finite difference approximation of Navier-Stokes equations, staggered grid arrangements are usually employed, because they possess several favourable properties due to a more  
Figure 8: Iso-contours of vorticity for levels \(-6,-4,\ldots,+6\) at time \(t=8\) for several schemes and formulations. In all cases, the skew-symmetric form of the convective term is employed and the classical RK4 scheme is used for time integration, unless otherwise specified. The first row shows results for explicit schemes of increasing accuracy on a collocated layout; the second row refers to classical compact schemes on a staggered layout; the third row to the novel Hermitian schemes on a staggered layout. Figure 9: Time evolution of kinetic energy for all the methods presented in Figure 8, which also serves as a reference for the labels. robust coupling between the discrete sets of variables. Moreover, when dealing with high Reynolds number flows, or in any case in which one is forced to a step size that is at the edge of (or even over) the acceptable resolution, high order and in particular compact schemes are possibly preferred for their improved resolving capabilities. Finally, in these circumstances the discrete approximation of the nonlinear convective term has to be performed by resorting to the so-called skew-symmetric form, in which two different expressions of the convective term are calculated and then averaged. This choice is practically unavoidable if one wants to prevent the nonlinear instability due to the accumulation of aliasing errors. These considerations show that staggered Navier-Stokes solvers based on compact schemes have several issues. They must compute different interpolations and staggered first derivatives of the velocity field, as well as collocated second derivatives for the approximation of the viscous term. The number of derivatives and interpolations required for each computational node is increased with respect to standard formulations, and each of these approximations needs different closure schemes near the boundaries. In this paper, a novel procedure to efficiently deal with these difficulties is presented. In this method, a whole set of different compact derivatives and interpolations is computed by explicit formulae resting on a single preliminarly computed compact approximation. Although a complete optimization of the application of the general theory to an incompressible Navier-Stokes solver has not been attempted, it has been shown that the procedure is cost effective and that conducts to novel compact schemes with interesting resolution properties. Moreover, it simplifies the treatment of boundary conditions, since only one boundary closure is required for the whole set of approximations. It is interesting to stress that although this paper has focused on the staggered discretization of incompressible flows, the proposed new method can be in principle equally applied to compressible flows. Numerical discretization of compressible flows has been usually carried out by considering regular or collocated arrangements of the variables. Staggered layouts have been considered as unnecessary, because of the absence of spurious modes for the pressure and in light of an increased programming difficulty. However, recently staggered discretizations have been proposed also for compressible flows, with the aim of alleviating the inherent instability of the simulations at high Reynolds numbers [17; 45]. In such situations compact schemes and skew-symmetric form of nonlinear convective terms are adopted, and the method here presented applies with slight modifications also to these situations. As a final comment, it is worth noting that, besides the application here proposed to staggered Navier-Stokes solvers, the theory developed is believed to have a remarkable interest by itself. It establishes that sets of compact approximations can be obtained by explicit evaluations based on the computation of a single compact scheme.