ABSTRACTThis paper addresses the temporal dynamics of the time marching when this approach is used for determining the steady solution of advection–diffusion problems. The time-step size, , is the main parameter and is the number of time steps which is needed to reach the steady solution, from a given initial state and according to a pre-defined time-convergence criterion. The measure of obtains by proceeding to march in time with a large set of values. Four distinct advection–diffusion problems are considered. They are time discretized by a usual second-order Finite-Difference scheme, and their spatial behaviour is computed by using two different spectral solvers. The space-grid parameter is N, the spectral cut-off for any space direction. A universal behaviour is observed, giving rise to V-shaped plots when the Courant-Friedrich-Levy instability imposes its control to the march-in-time. Moreover, one of the solvers leads to U-shaped plots for the incompressible-flow configurations in weakly nonlinear regimes. This is the temporal consequence of the time-splitting method that this solver has adopted for uncoupling the velocity from the pressure.