  We describe a new CSPH-TVD method for numerical integration of hydrodynamical
equations. The method is based on combined Lagrange-Euler approaches, and it
has been devoted to simulations of hydrodynamical flows in various
astrophysical systems with non-homogeneous gravitational fields and the
non-steady boundary between gas and vacuum. A numerical algorithm was tested on
analytical solutions for various problems, and a detailed comparison of our
method with the MUSCL scheme is also presented in the paper. It is shown that
the CSPH-TVD scheme has a second order of accuracy for smooth solutions
(well-balanced approach) and it provides reliable solutions in the vicinity of
strong shock waves and at the open gas-vacuum interfaces. We also study the
effectiveness of parallel implementations of CSPH-TVD method for various NVIDIA
Tesla K20/40/80, P100 graphics processors.
