On the efficiency of a numerical method with periodic time strides for solving incompressible flows E. Sanmiguel-Rojas, J. Ortega-Casanova, R. Fernandez-Feria * Universidad de Malaga, E.T.S.Ingenieros Industriales, 29013 Malaga, Spain Received 28 February 2002; received in revised form 11 December 2002; accepted 14 January 2003 Abstract An explicit numerical method to solve the unsteady incompressible flow equations consisting on N small time steps Dt between each two much larger time steps ðDtÞ 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 Dt slightly smaller than the critical one (DtÞ 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 ðDtÞ 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 ðDtÞ 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. Ó 2003 Elsevier Science B.V. All rights reserved. 1. Introduction Explicit numerical methods are sometimes preferred to implicit ones to solve the incompressible Navier–Stokes equations owing to the high computational cost of an implicit method at each time step, particularly for multi-dimensional flows at moderate and high Reynolds numbers. However, explicit methods have severe stability constrains over the time step, which usually has to be several orders of magnitude smaller than the time step one would like to use in accordance with the resolution of the spatial grid. These constraints are particularly severe for low-Reynolds-numbers flows (see, for example [1,2]). Journal of Computational Physics 186 (2003) 212–229 www.elsevier.com/locate/jcp * Corresponding author. 0021-9991/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0021-9991(03)00049-4