Galilean invariance and stabilized methods for compressible flows.

*(English)*Zbl 1207.76094Summary: In a recent work [the author et al., Comput. Methods Appl. Mech. Eng. 196, No. 4–6, 966–978 (2007; Zbl 1120.76334)], it was observed that lack of Galilean invariance led to catastrophic instabilities when stabilized methods were used in Lagrangian shock hydrodynamics computations. By means of an arbitrary Lagrangian-Eulerian (ALE) formulation, Galilean invariant SUPG operators were consistently derived in [the author, Comput. Methods Appl. Mech. Eng. 196, No. 4–6, 1108–1132 (2007; Zbl 1120.76333)], and their Lagrangian and Eulerian limits were compared to the most commonly used stabilized formulations. In the particular case of Eulerian meshes, it was shown that most of the SUPG operators designed to date for compressible flow computations are not invariant. However, due to the significant overhead of algebraic manipulations, the use in (the author, loc. cit.) of the referential form of the ALE equations made the presentation of the main ideas quite involved. The present paper addresses this particular issue, since the invariance analysis is presented with the aid of the intuitive current configuration reference frame, more familiar to computational fluid dynamicists.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

76N99 | Compressible fluids and gas dynamics |

##### Keywords:

variational multiscale methods; stabilized methods; SUPG methods; Galilean transformations; invariance
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\textit{G. Scovazzi}, Int. J. Numer. Methods Fluids 54, No. 6--8, 757--778 (2007; Zbl 1207.76094)

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