On the efficiency of a numerical method with periodic time strides for solving incompressible flows

An explicit numerical method to solve the unsteady incompressible flow equations consisting on N small time steps Δt between each two much larger time steps (Δt)1 is considered. The stability and efficiency of the method is first analyzed using the one-dimensional diffusion equation. It is shown that the use of a time step Δt slightly smaller than the critical one (Δt)c given by numerical stability allows to periodically take a much larger time step (stride) that speeds-up the advance in time in a numerical stable scheme. In particular, the stability analysis shows that for a given value of the stride (Δt)1, there is an optimum value of the small time step for which the computational speed is the fastest (N is a minimum), being this speed significantly larger than the corresponding one for an explicit method using (Δt)c only. The efficiency of the method is discussed for different time discretization schemes. The numerical method is then used to solve a particular incompressible flow. It is shown that the method is significantly (about three times) faster than a standard explicit scheme, and yields the same time evolution of the flow (within spatial accuracy). Further, it is shown that a much more higher computational speed and efficiency is reached if one combines an implicit scheme for the periodic strides with the explicit small time steps. With this combination one can speed-up the computations in more than one order of magnitude with the same resolution.