On $L^2$-dissipativity of a linearized difference scheme on staggered meshes with a quasi-hydrodynamic regularization for $\mathrm{1D}$ barotropic gas dynamics equations

We study an explicit two-level finite difference scheme on staggered meshes, with a quasi-hydrodynamic regularization, for $\mathrm{1D}$ barotropic gas dynamics equations. We derive necessary conditions and sufficient conditions close to each other for $L^2$-dissipativity of solutions to the Cauchy problem for its linearization on a constant solution, for any background Mach number $M$. 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 coefficient or the physical viscosity one is used. A comparison with the spectral von Neumann condition is also given for $M=0$.