A third-order multistep time discretization for a Chebyshev-tau method

A time discretization scheme based on the third-order backward difference formula has been embedded into a Chebyshev tau spectral method for the Navier–Stokes equations. The time discretization is a variant of the second-order backward scheme proposed by Krasnov et al. (2008) [3]. High-resolution direct numerical simulations of turbulent incompressible channel flow have been performed to compare the backward scheme to the Runge–Kutta scheme proposed by Spalart et al. (1991) [2]. It is shown that the Runge–Kutta scheme leads to a poor convergence of some third-order spatial derivatives in the direct vicinity of the wall, derivatives that represent the diffusion of wall-tangential vorticity. The convergence at the wall is shown to be significantly improved if the backward scheme is applied.