A projection-difference scheme for the unsteady motion of a viscous barotropic gas

A new implicit projection-difference scheme for the unsteady motion of a viscous barotropic gas is proposed in terms of Eulerian coordinates for the cases of one, two, and three spatial variables. The peculiarity of this scheme consists in the fact that we approximate the continuity equation written in terms of the logarithm of density, which allows one to ensure the positiveness of the density function for any parameters of the scheme. This scheme is two-layered. At each time step, a numerical solution is found by solving a linear system. The existence and uniqueness theorem for a numerical solution is proved without no additional conditions imposed on the time and spatial discretization parameters. It is shown that the scheme can be used in the problems with nonsmooth initial conditions in the one-dimensional case and in the cavity problem in the two-dimensional case. The importance of including the artificial viscosity in the approximation of the continuity equation is shown experimentally.