On $L^2$-dissipativity of a linearized scheme on staggered meshes with a regularization for 1D barotropic gas dynamics equations

We study an explicit two-level finite difference scheme on staggered meshes, with a quasi-hydrodynamic regularization, for 1D barotropic gas dynamics equations. We derive both necessary conditions and sufficient conditions close to them, for $L^2$-dissipativity of solutions to the Cauchy problem linearized on a constant solution, for any background Mach number $\mathrm{M}$. We apply the spectral approach and analyze matrix inequalities containing symbols of symmetric matrices of convective and regularizing terms. We consider the cases where either the artificial viscosity or the physical viscosity is used. A comparison with the spectral von Neumann condition is also given for $\mathrm{M}$.