$L^2$-dissipativity of finite-difference schemes for $\mathrm{1D}$ regularized barotropic gas dynamics equations at small Mach numbers

We study explicit two-level finite-difference schemes on staggered meshes for two known regularizations of $\mathrm{1D}$ barotropic gas dynamics equations including schemes with discretizations in $x$ that possess the dissipativity property with respect to the total energy. We derive criterions of $L^2$-dissipativity in the Cauchy problem for their linearizations at a constant solution with zero background velocity. We compare the criterions for schemes on non-staggered and staggered meshes. Also we consider the case of $\mathrm{1D}$ Navier–Stokes equations without artificial viscosity coefficient. For one of their regularizations, the maximal time step is guaranteed for the choice of the regularization parameter $\tau_{opt}=\nu_*/c^2_*$, where $c_*$ and $\nu_*$ are the background sound speed and kinematic viscosity; such a choice does not depend on the meshes. To analyze the case of the $\mathrm{1D}$ Navier–Stokes–Cahn–Hilliard equations, we derive and verify the criterions for $L^2$-dissipativity and stability for an explicit finite-difference scheme approximating a nonstationary $4^{\text{th}}$-order in $x$ equation that includes a $2^{\text{nd}}$-order term in $x$. The obtained criteria may be useful to compute flows at small Mach numbers.