Finite difference schemes for solving system equations of gas dynamic in a class of discontinuous functions

In this paper, the difference scheme for solving nonlinear system of equations of gas dynamic problems in a class of discontinuous functions is investigated. Firstly, for some simple cases, the nature of solution of the differential equations describing one-dimensional, constant pressure, and isentropic flow of compressible fluids are considered. It has been proved that the solution of this system equations has discontinuous points whose positions are unknown beforehand. In order to obtain the solution of the main problem in a class of discontinuous functions, the auxiliary problem is suggested. The degree of differentiability of the solution of the auxiliary problem is higher than the degree of differentiability of solution of the main problem. Furthermore, the suggested auxiliary problem provides to write out effective and higher order numerical algorithms. The solutions obtained from these algorithms represent the physical nature of the problem with a high accuracy. Some properties of numerical solution are also investigated. Additionally, some numerical experiments are carried out by means of using the auxiliary problem.

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