Multidimensional nodal method of characteristics for hyperbolic systems

Disclosed is a multidimensional nodal method of characteristics, designed to integrate hyperbolic systems, based on splitting the initial system of equations into a number of one-dimensional subsystems, for which a one-dimensional nodal method of characteristics is used. Calculation formulas are given, the calculation method is described in detail in relation to a single-speed model of a heterogeneous medium in the presence of gravity forces. The presented method is applicable to other hyperbolic systems of equations. Using this explicit, non-conservative, first-order accuracy of the method, a number of test tasks are calculated and it is shown that in the framework of the proposed approach, by attracting additional points in the circuit template, it is possible to carry out calculations with Courant numbers exceeding one. So, in the calculation of the flow of the three-dimensional step by the flow of a heterogeneous mixture, the Courant number was 1.2. If Godunov's method is used to solve the same problem, the maximum number of Courant, at which a stable account is possible, is $0.13\times10^{-2}$. Another feature of the multidimensional method of characteristics is the weak dependence of the time step on the dimension of the problem, which significantly expands the possibilities of this approach. Using this method, a number of problems were calculated that were previously considered “heavy” for the numerical methods of Godunov, Courant–Isaacson–Rees, which is due to the fact that it most fully uses the advantages of the characteristic representation of the system of equations.