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

We show that for the P-N-P-N-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 simulatio ns. These instabilities depend on the formulation of the convection operato r. The numerical scheme is stable for the convective form and one version o f the rotational form but unstable for the divergence form and the skew-sym metric 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 sta ggered grid between velocity and pressure, as often used in SEM. (C) 2000 I MACS. Published by Elsevier Science B.V. All rights reserved.