New generation computational algorithms for systems of hyperbolic equations

The system of equations of gas dynamics (Euler equations) consists of integral conservation laws, which in the areas of smoothness of the solution are reduced to a system of quasi-linear partial differential equations of hyperbolic type. Among the methods of numerical solution of these equations, the most widespread at present are difference schemes, which are based on the flow form of writing difference equations (finite volume method) and the problem of discontinuity decay as a method of calculating flows (S.K. Godunov, 1959). For the time that has passed since the publication of the basic methodology, method C.K. Godunov was significantly improved and generalized to other classes of problems described by conservation laws. In all these algorithms, known as "Godunov-type schemes", the hyperbolicity of conservation laws is hidden in the Hugonio ratios, or their simplified analogues (Ph. Roe), and the fact of the existence of characteristics, as the most significant sign of hyperbolicity, was not used in any way.
Another approach to the numerical solution of hyperbolic conservation law equations (CABARET scheme) is also based on the flow form of writing difference equations, however, the characteristic form of the original equations is used to calculate the flows, not the decay of the gap, but the characteristic form of the original equations. The flows are formed on the basis of local Riemann invariants (quasi-invariants) determined by the approximation of the characteristic shape within each space-time calculation cell. The term "CABARET scheme" is the designation of a whole class of conservatively characteristic (KX) difference schemes for systems of hyperbolic equations loaded (perturbed) by integral and differential operators of high orders. The CABARET scheme can be considered as an alternative to the method of S.K. Godunov.
The report is devoted to a brief overview of the history of the development of CCS, analysis of their features and examples of use in aeroacoustics, hydrogen safety and computational oceanology.