Performance of high‐order schemes on collocated and staggered grids

Abstract

Direct numerical simulation of turbulent flows requires tremendous amount of computational power because it is necessary to resolve the spatial structures of the flow down to the Kolmogorov length scale. This procedure generates the socalled grids or meshes. There are two competing grids favour, namely a collocated and staggered grid. To date, it is unclear which one is superior in terms of the resolution power. Recent advance in numerical methods for solving turbulent flows indicates that a staggered grid could be a better candidate for solving turbulent flows due to its superiority in mass-conservation. In order to clarify this possibility, the accuracy of these two grids must be tested using an identical numerical method. In this work, we use a pseudo-spectral method to mimic the behaviour of each numerical approximation. The effects of numerical errors are represented by the modified wavenumber which is then converted back to the physical space. The numerical methods considered in this work are the second-, fourth- and sixth-order central-finite-difference approximations. First a priori error analysis will be investigated, followed by a posteriori error analysis and cross-correlation against the reference solution from the pseudospectral scheme. It is found that the second-order scheme on a staggered grid is as good as the fourth-order scheme on a collocated grid which costs twice more expensive.