Splitting Techniques for the Unsteady Stokes Equations

A pseudospectral (or collocation) approximation of the unsteady Stokes equations is presented. Using the Uzawa algorithm the spectral system is decoupled into Helmholtz equations for the velocity components and an equation with the pseudo-Laplacian for the pressure. Both the velocity components and the pressure are approximated by polynomials of the same degree N. Compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed. The resulting discrete velocity field is identically divergence-free. Only one grid (no staggered grids) with the standard Chebyshev Gauss--Lobatto nodes is used. Here we further compare our treatment with an inhomogeneous Neumann condition for the pressure. This scheme was proposed in [G. Karniadakis, M. Israeli, and S. A. Orszag, J. Comp. Phys., 97 (1991), pp. 414--443]. The comparison shows that the Neumann condition yields somewhat better approximations for the pressure. In the time discretization, a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure. It is numerically shown that a stable third order method in time can be achieved. In the case of periodic boundary conditions this result is confirmed by Fourier analysis.