On a Godunov-type numerical scheme for describing the gas and dust components in problems of star formation

The paper presents one construction of the Godunov-type method based on the separation of operators describing the work of pressure forces and advective transfer. Separate consideration of advective transfer makes it possible to describe the motion of both gas and dust components within the framework of a single numerical scheme. In the case of describing gas dynamics, the work of pressure forces is taken into account at a separate stage, regardless of transfer. This makes it possible to use the numerical scheme in solving star formation problems, where it is necessary to jointly solve the equations of hydrodynamics and equations for dust motion. A piecewise parabolic representation of physical variables in all directions is used to reduce the dissipation of the numerical method. The numerical method has been verified on the Riemann problems for a hydrodynamic and dust discontinuity, the Sedov problem of point explosion, and the problem of dust cloud collapse, which have an analytical solution.