  In this paper we consider various splitting schemes for unsteady problems
containing the grad-div operator. The fully implicit discretization of such
problems would yield at each time step a linear problem that couples all
components of the solution vector. In this paper we discuss various
possibilities to decouple the equations for the different components that
result in unconditionally stable schemes. If the spatial discretization uses
Cartesian grids, the resulting schemes are Locally One Dimensional (LOD). The
stability analysis of these schemes is based on the general stability theory of
additive operator-difference schemes developed by Samarskii and his
collaborators. The results of the theoretical analysis are illustrated on a 2D
numerical example with a smooth manufactured solution.
