2023 , Volume 28, № 2, p.58-71

We consider detailed description of the quasi-invariants method for constructing a CABARET scheme approximating a hyperbolic system of conservation laws that cannot be written in the form of invariants. A classification of quasi-invariants with respect to their non-linear dependence on the desired functions for the corresponding non-divergent notation of the approximated system and describe a method for obtaining quasi-invariants of a given order of nonlinearity is presented. The algorithm of the resulting CABARET scheme is given for the unidirectional case when in the speed of the calculated exact solution the characteristics of one family does not change sign. As a specific example, we have considered the conservation laws system of non-isoentropic gas dynamics that admits three different families of quasi-invariants obtained from the classical nondivergent form of this system for the density, velocity, and entropy functions. Each family of these invariants corresponds to its own modification of the CABARET scheme. A comparative analysis for the accuracy of these modifications applied for calculation of the Sod problem on the initial discontinuity decay in a polytrophic gas is carried out. On the basis of this analysis, the optimal form of quasi-invariants has been identified, which allows localizing strong and weak discontinuities of the exact solution with high accuracy using the CABARET scheme.