Stability Analysis for Different Formulations of the Nonlinear Term in PN−PN−2 Spectral Element Discretizations of the Navier–Stokes Equations

We show that for the PN−PN−2 spectral element method, in which velocity and pressure are approximated by polynomials of order N and N−2, respectively, numerical instabilities can occur in the spatially discretized Navier–Stokes equations. Both a staggered and nonstaggered arrangement of the N−2 pressure points are considered. These instabilities can be masked by viscous damping at low Reynolds numbers. We demonstrate that the instabilities depend on the formulation of the nonlinear term. The numerical discretization is stable for the convective form but unstable for the divergence and the skew-symmetric form. Further numerical analysis indicates that this instability is not caused by nonlinear effects, since it occurs for linearized systems as well. An eigenvalue analysis of the fully discretized system shows that an instability is introduced by the formulation of the nonlinear term. We demonstrate that the instability is related to the divergence error of the computed solution at those velocity points at which the continuity equation is not enforced.