Combined compact difference method for solving the incompressible Navier-Stokes equations

This paper presents a numerical method for solving the two-dimensional unsteady incompressible Navier–Stokes equations in a vorticity–velocity formulation. The method is applicable for simulating the nonlinear wave interaction in a two-dimensional boundary layer flow. It is based on combined compact difference schemes of up to 12th order for discretization of the spatial derivatives on equidistant grids and a fourth-order five- to six-alternating-stage Runge–Kutta method for temporal integration. The spatial and temporal schemes are optimized together for the first derivative in a downstream direction to achieve a better spectral resolution. In this method, the dispersion and dissipation errors have been minimized to simulate physical waves accurately. At the same time, the schemes can efficiently suppress numerical grid-mesh oscillations. The results of test calculations on coarse grids are in good agreement with the linear stability theory and comparable with other works. The accuracy and the efficiency of the current code indicate its potential to be extended to three-dimensional cases in which full boundary layer transition happens