Stability of Implicit Methods on Staggered Grids

Abstract

The numerical solution of the Navier-Stokes equations with free surface when the Reynolds number is very small (Re 1) requires the use of implicit time discretization. The parabolic stability condition of explicit methods imposes severe restrictions on the time-step making them too time consuming to be useful in practice. In the context of staggered grids, this work presents a study of the numerical stability of implicit methods. This stability is directly connected to the appropriate use of boundary conditions on the free surface and on rigid walls. The boundary conditions must be discretized with care so that the resulting method does not become conditionally stable. The stability results are derived for the model problem of the heat equation, and then applied to Navier-Stokes equations examples.