Stable and unstable formulations of the convection operator in spectral element simulations Original Research Article

We show that for the PN–PN−2 spectral element method (SEM), in which the velocity and pressure are approximated by polynomials of order N and N−2, respectively, numerical instabilities may occur in Navier–Stokes simulations. These instabilities depend on the formulation of the convection operator. The numerical scheme is stable for the convective form and one version of the rotational form but unstable for the divergence form and the skew-symmetric form. Further numerical analysis indicates that this instability is not caused by nonlinear effects but occurs also for the linearized momentum equations. We demonstrate that the instability is a consequence of the staggered grid between velocity and pressure, as often used in SEM.