Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

Emmanuel Franck (1, 2) , Laurent Gosse (3)

1 IRMA - Institut de Recherche Mathématique Avancée
2 TONUS - TOkamaks and NUmerical Simulations
3 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a time-marching, Lax–Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the Hs norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former “shape functions” and “symmetric potential schemes” are highlighted.

Mots clés: 2D Lax- Wendroff scheme; Linearized Euler system; Kirchhoff exact solution; Numerical vorticity dissipation

Domaines: Analyse numérique [math.NA]

Emmanuel Franck, Laurent Gosse. Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems. Annali dell'Universita di Ferrara, 2017, pp.1-26. ⟨10.1007/s11565-017-0296-9⟩. ⟨hal-01666060⟩