Finite difference schemes for hyperbolic conservation rules in the class of discontinuous functions and its application to the gas dynamics problem

The finite difference method in a class of discontinuous functions for obtaining the entropy solution of the Riemann problem of nonlinear hyperbolic equation having more than one lines of discontinuity in initial profile which describing of the 2D conservation rules is proposed. For this purpose, an auxiliary problem, which has advantages not found in the main problem and is equivalent to the main problem in a certain sense, is proposed. Using the advantages of the proposed auxiliary problem, the finite difference method with high precision is developed. By using the solution of the auxiliary problem, the solution that expresses the physical structure of the main problem properly has been obtained. With the help of the proposed method, the mutual effect of the waves has also been studied.