Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier–Stokes equations

This article presents a family of very high-order non-uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible Navier–Stokes equations. The high-order compact schemes on non-uniform grids developed in Shukla and Zhong [R.K. Shukla, X. Zhong, Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys. 204 (2005) 404] for linear model equations are extended to the full Navier–Stokes equations in the vorticity and streamfunction formulation. Two methods for the solution of Helmholtz and Poisson equations using high-order compact schemes on non-uniform grids are developed. The schemes are constructed so that they maintain a high-order of accuracy not only in the interior but also at the boundary. Second-order semi-implicit temporal discretization is achieved through an implicit Backward Differentiation scheme for the linear viscous terms and an explicit Adam–Bashforth scheme for the non-linear convective terms. The boundary values of vorticity are determined using an influence matrix technique. The resulting discretized system with boundary closures of the same high-order as the interior is shown to be stable, when applied to the two-dimensional incompressible Navier–Stokes equations, provided enough grid points are clustered at the boundary. The resolution characteristics of the high-order compact finite difference schemes are illustrated through their application to the one-dimensional linear wave equation and the two-dimensional driven cavity flow. Comparisons with the benchmark solutions for the two-dimensional driven cavity flow, thermal convection in a square box and flow past an impulsively started cylinder show that the high-order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary.