On the correctness of numerical simulation of gravitational instability with the evolution of multiple gravitational collapses

We study the correctness of numerical models based on the SPH-method for nonstationary problems of gravitational gas dynamics with the development of gravitational instability, including with the formation of multiple collapses of a gas in a circumstellar disk. The original differential initial boundary value problems are ill-posed because of their instability with respect to variations of input data. It is shown that the following regularization is performed in the SPH-based numerical method of solving such problems: (1) if the solution to the original unstable problem exists on the entire time axis, then the condition of the method's stability with respect to small variations of input data is satisfied due to a bounded range of the variables; (2) if the solution of the original unstable problem exists only on a bounded time interval, as in the case of multiple collapses, then a stable numerical method can be developed on a class of functions bounded by a fixed constant chosen from physical considerations. On this class of bounded functions, the original problem becomes well-posed. A combination of the SPH method with a grid-based method of calculating the gravitational forces allows one to ensure the boundedness of numerical solutions. In order to clarify a meaning of the approximate numerical solutions obtained by computer simulations, it is necessary to use the integral functions weakly sensitive to the details of the numerical algorithm in use.