An algorithm for solving transient problems of gravitational gas dynamics: a combination of the SPH method with a grid method of gravitational potential computation

A new numerical algorithm to solve the unsteady equations of gravitational gas dynamics in the thin disk approximation is proposed. This algorithm is based on a combination of the meshless SPH (Smoothed Particle Hydrodynamics) method for gas dynamics and the convolution method for solving Poisson's equation on a Cartesian grid. This convolution method is of high performance due to the fact that the grid potential function is computed and stored only in the plane of the disk. The efficiency of the algorithm is demonstrated by numerical experiments on the formation of structures in a circumstellar disk. We compare the results obtained by using the grid method for solving Poisson's equation in Cartesian and cylindrical geometry and show that in both these cases it is possible to reproduce the solutions with axial symmetry and to illustrate the formation of solitary regions of enhanced density.