Generalized and variational statements of the Riemann problem with applications to the development of Godunov's method

This paper is devoted to the numerical method for solving the fluid dynamics equations proposed by Godunov more than 60 years ago. This method is an explicit finite volume first-order discretization of the system of balance differential equations with the approximation of the numerical flux on the faces of the computational cells based on the exact solution of the Riemann problem. Statements of the generalized and variational Riemann problem are considered, and it is shown how the solutions to these problems can be used for the development of Godunov's method. In particular, the issues of increasing the order of approximation, constructing an implicit–explicit stable temporal scheme that is reduced to the second-order Godunov scheme when only the purely explicit component is used, solving discrete equations of the implicit–explicit scheme using the solution to the variational Riemann problem, and a generalization of Godunov's method for the linearized system of Euler equations are considered. Approximate solutions to the generalized Riemann problem that are analogs of the HLL and HLLC solutions for the case of linear initial data are obtained.