High-Order Direct Stokes Solvers with or Without Temporal Splitting: Numerical Investigations of Their Comparative Properties

A recently proposed direct Stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed, in comparison, first with the splitting scheme proposed by G. Karniadakis, M. Israeli, and S. Orszag in [ J. Comput. Phys., 97 (1991), pp. 414--443], and with the unique grid (${\Bbb{P}}_{N}, {\Bbb{P}}_{N-2}$) Uzawa approach for the space accuracy and computational costs. The Chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators, and their discrete time versions, from the first to the fourth order. An explicit boundary condition is also involved in the proposed Stokes solver, and it is numerically shown that the trace of ${\bf \Delta }{ \bf u}$ on the boundary must be evaluated through its $-{\bf \nabla} \times {\bf \nabla} \times$ contribution only; otherwise the ellipticity is lost before proceeding to the time discretization. The explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable, while the schemes built at the next two higher orders are limited by the usual explicit (${\cal {O}}(N^{-4})$) stability criterion on the time step. The proposed $({\bf u},p)$ decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes. The effective accuracy obtained for the Navier--Stokes 2D solutions has been measured, for the Re=1000 lid-driven cavity problem, and has been found equivalent to what is supplied by the much more expensive Uzawa decoupling method.