On Conditions for L2-Dissipativity of Linearized Explicit QGD Finite-Difference Schemes for One-Dimensional Gas Dynamics Equations

An explicit two-level in time and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L²-dissipativity of solutions to the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity and depending on the Mach number. It is shown that the condition on the Courant number can be made independent of the Mach number. The results provide a substantial development of the well-known stability analysis of the linearized Lax–Wendroff scheme.