Finite volume discretization and multilevel methods in flow problems

This article is intended as a preliminary report on the implementation of a finite volume multilevel scheme for the discretization of the incompressible Navier-Stokes equations. As is well known the use of staggered grids (e.g. MAC grids, Perićet al. Comput. Fluids,16(4), 389–403, (1988)) is a serious impediment for the implementation of multilevel schemes in the context of finite differences. This difficulty is circumvented here by the use of a colocated finite volume discretization (Faureet al. (2004a) Submitted, Perićet al. Comput. Fluids,16(4), 389–403, (1988)), for which the algebra of multilevel methods is much simpler than in the context of MAC type finite differences. The general ideas and the numerical simulations are presented in this article in the simplified context of a two-dimensional Burgers equations; the two-, and three-dimensional Navier-Stokes equations introducing new difficulties related to the incompressibility condition and the time discretization, will be considered elsewhere (see Faureet al. (2004a) Submitted and Faureet al. (2004b), in preparation).