Theoretical and Numerical Analysis of One Euler Two-Layer Completely Conservative Difference Scheme of Gas Dynamics with Adaptive Viscosity

For the equations of gas dynamics in Euler variables, using the operator approach, a family of two-layer in time completely conservative difference schemes (CCDS) with space-profiled weight factors used to approximate the momentum and energy fluxes in time is constructed, theoretically and numerically studied. The schemes have second-order accuracy and are implemented using simple iterative processes. A class of divergent adaptive viscosities for these CCDS is developed and a theoretical analysis of their stability is carried out. The regularization of flux terms of the gas dynamics equations with the help of adaptive artificial viscosity is proposed and, using the example of well-known Einfeld problem, is numerically investigated. This regularization effectively eliminates nonphysical oscillations of the solution, entropy peaks and preserves the property of complete conservatism of schemes of this class.