L^2-Dissipativity of Difference Schemes for Regularized 1D Barotropic Gas Dynamic Equations at Low Mach Numbers

Mathematical Models and Computer Simulations. 2021. Vol. 13. No. 6. P. 1097–1108.

Zlotnik A.A., Lomonosov T.A.

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.

Keywords: explicit finite-difference scheme, 1D gas dynamics equations, L^2-dissipativity, staggered meshes, Navier-Stokes-Cahn-Hilliard equations