The authors, in a previous paper [1], noticed large errors in turbulence dissipation rate and enstrophy near the wall of a channel when solving turbulent channel flow using a spectral Chebyshev tau method in space and a Runge-Kutta (RK) in time. The time discretization is third order for the convective and second-order for the viscous terms.

In this paper, the authors discuss a third-order variant of the second-order backward time discretization method. They analyze the stability of the method and compare the numerical solution of a turbulent channel flow to previous results using RK and using Crank-Nicolson Adams-Bashforth. The new method requires double the storage used by RK. It was demonstrated numerically that the near-wall errors dropped by more than six orders of magnitude when using a third-order backward scheme with 512 Chebyshev modes in the wall normal direction. This drop did not occur when using RK.

The third-order multi-step method for time discretization developed here is not new, but it was shown to reduce the errors near the wall in solving Navier-Stokes equations in bounded domains.