# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 1 Introduction
In this last paper also the problem of boundary conditions is investigated, and a series of asymmetric formulae for the boundary is presented. In these early years the focus was not only on the applications of compact schemes, but also on the development of a theoretical framework in which new schemes could be derived. In a series of papers, [PERSON] and co-workers [8; 9; 10] firstly established that many compact schemes can be derived by employing the theory of spline interpolation. They showed, for example, that the classical Pade fourth-order formula for first derivative can be obtained by analytically differentiating the cubic spline interpolation, thus recognizing that compact schemes could be derived through a theory of polynomial interpolation in physical space, which has to be necessarily implicit. They also obtained formulae for second derivatives and for the case of nonuniform mesh, and applied their schemes to a variety of fluid flow problems. [PERSON] et al. [11] developed the theory of the \"Operator Compact Implicit Method\", in which new classes of compact schemes were derived for parabolic problems by applying the compact technique to the full spatial convection-diffusion operator. [PERSON] [12], while investigating the problem of boundary condition closures, proposed also a new method for obtaining a second derivative approximation for interior points through the so-called \"explicit elimination\", in which the second derivative is expressed as a function of the values and of the computed set of first derivatives, in a way which shares some similarities with the technique adopted in this paper. Finally, [PERSON] and [PERSON] [13] introduced for the first time the formulae for compact interpolation and for compact staggered derivatives in their applications to geostrophic adjustment problem and to the nonlinear two-dimensional vortex-advection problem. In the first application, they also compared the various schemes by computing the discrete dispersion relation of the gravity waves, in an analysis which is practically equivalent to the Fourier analysis of the modified wavenumber, which is now a standard tool for the evaluation of the resolution properties of difference schemes. The polynomial approach for the derivation of compact schemes has been pursued in further occasions during the past two decades. [PERSON] and [PERSON] [14; 15] used the theory of Hermite interpolants, based on first and second derivative data, to obtain the expressions for their so-called CCD schemes, in which both first and second derivatives appear as unknowns in the generic scheme. [PERSON] and [PERSON] [16] developed a new theory of Local Matched Reconstructions, which generalizes the spline interpolation theory and is more suited to the systematic generation of all the classical collocated compact schemes. In that paper the authors employ this theory also to derive new classes of compact schemes. In this paper, we present a strategy for the generation of several compact derivatives starting from the computation of a single set of compact schemes. The idea is motivated by the growing interest in the application of compact schemes to Navier-Stokes equations [17; 18; 19; 20; 21] and is here applied to a typical incompressible Navier-Stokes solver with a staggered arrangement of the variables. The convective term is discretized by using the skew-symmetric form, which is a mandatory choice for energy-stable simulations in the context of compact-difference schemes. In such a situation, the computation of convective and diffusive terms in the the momentum equation requires the evaluation of staggered derivatives and interpolations of velocity components, together with the collocated second derivative. It will be shown that an efficient and economic way of obtaining all these quantities can be devised, which is based on the computation of a single set of compact derivatives, i.e., through the solution of a single tridiagonal system. The paper is organized as follows.