Investigation of the Stability and Convergence of Difference Schemes for a Polytropic Gas with Subsonic Flows

We analyze the stability with respect to the initial data and the convergence in the uniform norm of difference schemes approximating the equations of a polytropic gas in terms of the Riemann invariants. We obtain conditions on the initial data providing the presence of only subsonic flows and the absence of shock waves in the medium in the course of time. We discuss the relationship between the notions of stability and monotonicity of difference schemes for nonlinear problems. We present the results of a numerical experiment that justify the obtained theoretical conclusions.