Numerical Evaluation of the Accuracy and Stability Properties of High-order Direct Stokes Solvers with or without Temporal Splitting

The temporal stability and effective order of two different direct high-order Stokes solvers are examined. Both solvers start from the primitive variables formulation of the Stokes problem, but are distinct by the numerical uncoupling they apply on the Stokes operator. One of these solvers introduces an intermediate divergence free velocity for performing a temporal splitting [KARNIA] while the other treats the whole Stokes problem through the evaluation of a divergence free acceleration field [BAKHLA],[LELA00]. The second uncoupling is known to be consistent with the harmonicity of the pressure field \cite{kn:LELA00}. Both solvers proceed by two steps, a pressure evaluation based on an extrapolated in time (of theoretical order $J_e$) Neumann condition, and a time implicit (of theoretical order $J_i$) diffusion step for the final velocity. These solvers are implemented with a Chebyshev mono-domain and a Legendre spectral element collocation schemes. The numerical stability of these four options is investigated for the sixteen combinations of $(J_e,J_i)$, $1 \leq J_e, J_i \leq 4$.