L^2-Dissipativity of Difference Schemes for Regularized 1D Barotropic Gas Dynamic Equations at Low Mach Numbers
Explicit two-level difference schemes on staggered meshes are studied for two well-known
regularizations of 1D barotropic gas dynamics equtions, including schemes with discretization in
x with the property of total energy dissipation. The criteria of L^2-dissipativity in the Cauchy problem
are derived for their linearizations on a constant solution with zero background velocity. The criteria for
schemes on nonstaggered and staggered meshes are compared. The case of 1D Navier-Stokes equations
without an artificial viscosity coefficient is also considered. For one of their regularizations, the
maximum time step is guaranteed by the choice of the regularization parameter $\tau_*=\nu_*/c_*^2$, where $c_*$ and $\nu_*$ are the background sound speed and kinematic viscosity; this choice is independent of the meshes.
To analyze the case of 1D Navier-Stokes-Cahn-Hilliard equations, the criteria are derived
and tested for the L^2-dissipativity and stability of an explicit difference scheme for a nonstationary equation
of the 4th order in x with a second order term in x. The obtained criteria can be useful in calculating
flows at low Mach numbers.