Stability of projection methods for incompressible flows using high order pressure-velocity pairs of same degree: Continuous and Discontinuous Galerkin formulations

E. Ferrer, D. Moxey, S. J. Sherwin, R. H. J. Willden

Commun. Comp. Phys., vol. 16, pp. 817-840 (2014)

@article{ferrer-2014,
  title = {{Stability of projection methods for incompressible flows using high order pressure-velocity pairs of same degree: Continuous and Discontinuous Galerkin formulations}},
  author = {Ferrer, E. and Moxey, D. and Sherwin, S. J. and Willden, R. H. J.},
  volume = {16},
  number = {3},
  pages = {817-840},
  doi = {10.4208/cicp.290114.170414a},
  year = {2014},
  journal = cicp,
  url = {https://davidmoxey.uk/assets/pubs/2014-temporal.pdf},
  abstract = {This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable, when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend 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 simulations when small time steps and low kinematic viscosities are used. Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global 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.}
}

Projection methods for incompressible flow become unstable below a certain time step, which is the opposite of the usual restriction and so easy to run into unexpectedly. This paper derives limits for that behaviour when velocity and pressure are approximated at the same polynomial degree, in both continuous and discontinuous Galerkin formulations, and shows the limit depends on the product of time step and kinematic viscosity rather than on the time step alone.