Two-Layer 1D Completely Conservative Difference Schemes of Gas Dynamics with Adaptive Regularization

摘要

Abstract The problem of the numerical solution of the system of one-dimensional equations of gas dynamics in Euler variables is considered. Despite the numerous known difference schemes for solving these equations, there are cases in which the 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 study is to construct a new nonlinear fully conservative difference scheme of the second order of approximation and accuracy in space and time, free from these disadvantages. The scheme proposed in the work is based on the scheme of A.A. Samarsky and Yu.P. Popov, but a-dditionally uses regularizing additives in the form of the adaptive artificial viscosity proposed by I.V. Fryazinov. The scheme is implicit in time and is implemented using the method of successive approximations. The stability conditions for the solution are theoretically obtained for it. The scheme has been tested on the Einfeldt problem and shock wave calculations. The results of the numerical experiments confirmed the necessary declared properties, i.e., the second order in space and time, complete conservatism, and monotonicity of the solution in the appropriate cases.