When solving the linearized Euler (LE) and linearized Navier-Stokes (LNS) equations using the finite element method (the standard Galerkin formulation). It can be shown that the method looses its good approximation characteristics when convective terms are present and when these terms locally dominate (Ref. 19). This can lead to spurious numerical oscillations. To remedy these oscillations stabilization is used.

The implementation of the stabilization methods follows the one discussed in Ref. 19 and Ref. 21. A general overview about stabilization methods can be found in Ref. 22. The GLS method combines stability and accuracy and the method order of accuracy is O(h2p+1) where p is the shape function order and h is the local mesh size. The SUPG method has an accuracy order of O(hp+1/2) for convection dominated problems and O(hp+1) for diffusion dominated problems.

In the linearized Euler interface the stabilization scheme implements the streamline upwind Petrov-Galerkin (SUPG) formulation of the weak form equations used for the finite element method. The stabilization constant αstab can be tuned depending on the problem solved, the nature of the background mean flow, and on the computational mesh. The implementation follows the one discussed in Ref. 10 and Ref. 14.