Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations (Communications in Computational Physics, 2014)
This paper presents limits for stability of projection type schemes whenusing high order pressure-velocity pairs of same degree. Two high order $h/p$ variational methods encompassing continuous and discontinuous Galerkin formulationsare used to explain previously observed lower limits on the time step for projectiontype schemes to be stable [18], when $h$- or $p$-refinement strategies are considered. Inaddition, the analysis included in this work shows that these stability limits depend not only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows.We show that high order methods prove advantageous in stabilising the simulationswhen small time steps and low kinematic viscosities are used.Drawing upon this analysis, we demonstrate how the effects of this instability canbe reduced in the discontinuous scheme by introducing a stabilisation term into theglobal system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.