# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 1 Introduction  
Direct numerical simulation of many complex multi-scale problems, such as turbulence in fluids, requires the use of adequately refined grids to resolve the structure and the dynamics of the smallest scales. However, in many situations the limits of available computing power, both in terms of memory requirements and of integration time, force the choice of the grid to a step size which is at (or even over) the edge of the acceptable resolution. In these situations the physics is described within a few computational nodes and the resolution properties of the discretization are a crucial factor. A similar situation is encountered in Large Eddy Simulation (LES) of turbulence. In this case, the smallest scale resolved by the discretization is deliberately greater than the smallest physical scales having significant energy, since the effects of the latter are modelled through the specification of a suitable subgrid model. In this case, the resolution of the discretization (intended as the number of mesh points required for a given accuracy at a given wave length) is even more important, since the numerical errors have to be small as compared to the subgrid energy contribution. In these and in many other cases, high-order methods are usually recommended, since they generally provide lower levels of numerical errors for the grids employed. In recent years high order implicit compact schemes have been extensively used for flow problems, as an alternative to high order explicit finite-difference schemes and to spectral (or pseudospectral) schemes. After the 1992 seminal paper by [PERSON] [1], in which a comprehensive and systematic presentation of compact schemes is reported, they have been extensively used in fluid flow calculations, both in compressible and incompressible cases. Their success is due to several favourable properties which place them in between the flexibility typical of explicit finite difference schemes and the resolution properties of global spectral schemes. Although the influential paper by [PERSON] is considered as the starting point for the subsequent huge research activity on compact schemes and on their applications, there is a quite interesting production on this subject well before the nineties, whose character is, in some sense, in line with theapproaches taken in this paper. Actually, implicit compact derivatives were already used in numerical analysis well before the seventies (cfr. for example [2]); however, their main application was limited to the finite-difference approximation of ordinary differential equations. The fourth-order implicit Nystrom formula, known as Simpson's rule, can indeed be viewed as an application of the classical Pade scheme to a system of first order ordinary differential equations. The first mention in the open literature of the opportunity of using implicit compact formulae for the numerical approximation of partial differential equations seems to be due to [PERSON] and [PERSON] [3], who attribute to [PERSON] the suggestion. In that paper, they fully recognized the main favorable characteristics of compact schemes, namely the reduced truncation error (with respect to the analogous explicit formulae) the reduced width of the stencil, which eases the implementation of boundary conditions, and the low computational cost. The idea of using implicit formulae for the computation of spatial derivatives can be possibly traced back to the finite difference formulation of some finite element methods based on Hermite cubics, which had growing popularity at that time (see the interesting observation made at p. 59 of ref. [4]). The authoritative suggestion contained in the review of [PERSON] and [PERSON] sparked many studies on the development and test of compact schemes. Almost simultaneously compact schemes were applied with success to the wave equation [5], to parabolic equations [6] and to different nonlinear fluid mechanics problems [7].