A Compact Finite Difference Method on Staggered Grid for Navier-Stokes Flows

Compact finite difference methods feature high-order accuracy with smaller stencils and easier application of boundary conditions, and have been employed as an alternative to spectral methods in direct numerical simulation and large eddy simulation of turbulence. The underpinning idea of the method is to cancel lower-order errors by treating spatial Taylor expansions implicitly. Recently, some attention has been paid to conservative compact finite volume methods on staggered grid, but there is a concern about the order of accuracy after replacing cell surface integrals by average values calculated at centers of cell surfaces. Here we introduce a high-order compact finite difference method on staggered grid, without taking integration by parts. The method is implemented and assessed for an incompressible shear-driven cavity flow at Re = 10, a temporally periodic flow at Re = 10, and a spatially periodic flow at Re = 10. The results demonstrate the success of the method.