"On the correctness of numerical simulation of gravitational instability
with the evolution of multiple gravitational collapses" Snytnikov V.N. and Stoyanovskaya O.P. 
We study the correctness of numerical models based on the SPHmethod 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 illposed because of their instability with respect to variations of input data. It is shown that the following regularization is performed in the SPHbased 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 wellposed. A combination of the SPH method with a gridbased 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. Keywords: wellposedness, numerical methods, gravitational gas dynamics, instability, stability with respect to the input data, stochastics, SmoothedParticle Hydrodynamics (SPH), circumstellar disc, gravitational collapse, gas clump.

