On Conditions for Weak Conservativeness of Regularized Explicit Finite-Difference Schemes for 1D Barotropic Gas Dynamics Equations

Abstract

We consider explicit two-level three-point in space finite-difference schemes for solving 1D barotropic gas dynamics equations. The schemes are based on special quasi-gasdynamic and quasi-hydrodynamic regularizations of the system. We linearize the schemes on a constant solution and derive the von Neumann type necessary condition and a CFL type criterion (necessary and sufficient condition) for weak conservativeness in L 2 for the corresponding initial-value problem on the whole line. The criterion is essentially narrower than the necessary condition and wider than a sufficient one obtained recently in a particular case; moreover, it corresponds most well to numerical results for the original gas dynamics system.

On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

“…For the QHD regularization, these 2D and 3D results are derived for the first time and, for the QGD regularization, they improve those that have recently been obtained in [29]. Previously, the L 2 -dissipativity analysis of similar schemes in the much simpler 1D barotropic case was accomplished for zero and any Mach number in [30][31][32], respectively. In this paper, we base on the papers in [29,32] and aim to develop further their technique.…”

“…where b (0) = min 1 k n b k and b (1) = 1. As max 1 k n q(|M k |) max 0 m M q(M), for = 0, condition (31) is more sharp than the second condition (37), whereas for = 1 they coincide.…”