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

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linearwave 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.