Two-layer 1D completely conservative difference schemes of gas dynamics with adaptive regularization

The problem of numerical solution of the system of one-dimensional equations of gas dynamics in Euler variables is considered. Despite the abundance of known difference schemes for solving these equations, there are cases in which standard methods are ineffective. For example, most of the known schemes do not resolve well the solution profiles in the Einfeldt problem and similar ones. Therefore, the aim of this work was to construct a new nonlinear completely conservative difference scheme of the second order of approximation and accuracy in space and time, free from the above disadvantages. The scheme proposed in the work is based on the scheme of A.A. Samarsky and Yu.P. Popov, but additionally uses regularizing additives in the form of adaptive artificial viscosity proposed by I.V. Fryazinov. The scheme is implicit in time and is implemented using the method of successive approximations. For it, the conditions for the stability of the solution are theoretically obtained. The scheme has been tested on the Einfeldt problem and shock wave calculations. The results of numerical experiments confirmed the necessary declared properties, namely, the second order in space and time, complete conservatism, monotonicity of the solution in appropriate cases.