Application of the CABARET Scheme for Calculating Multidimensional Hyperbolic Systems of Conservation Laws

The method of quasi-invariants is presented for constructing the standard and modified CABARET and CABARETM schemes that approximate the Cauchy problem for a multidimensional quasilinear hyperbolic system of conservation laws. As a specific example, the problem of a cylindrical explosion is considered, which is a two-dimensional Cauchy problem for the equations of nonisentropic gas dynamics with a polytropic equation of state and with special circular piecewise constant initial data. Test calculations have shown that the CABARET scheme is unstable when calculating this problem, while the numerical solution obtained by the CABARETM scheme is stable over a wide range of Courant numbers and localizes all the singularities of the approximated exact solution with high accuracy.