Full approximate storage (FAS) multigrid algorithm is the most commonly used multigrid algorithm for non-linear equations. The algorithm initially developed for steady-state equations was later extended to obtain steady-state solutions employing unsteady equations. In extending the FAS algorithm for the steady-state non-linear equations to unsteady non-linear equations, the FAS algorithm does not to take into account that the governing equations are typically linearized in time before they are solved. Thus, there is a scope to develop a new multigrid algorithm to apply the multigrid technique to the equations linearized in time. In the present work, such an algorithm is developed exploring this possibility and is implemented for two-dimensional incompressible and compressible flows coupled with explicit time marching procedures. The results of the new algorithm compare favourably with those of the FAS multigrid method and single grid. © 2002 Elsevier Science Ltd. All rights reserved.