About solvability and numerical simulation of nonstationary flow of incompressible fluid with a free surface

R.V. Shamin

P.P. Shirshov Institute of Oceanology of the Russian Academy of Sciences,

Moscow, Russia

We consider Dyachenko's equations describing nonstationary motion of ideal liquid with free boundary in a gravitational field. Dyachenko's equations are nonlinear integro-differential equations. They turn out to be convenient for numerical modeling.

Existence of analytic solutions of the above equations for a sufficiently small time interval is proved. Solutions from Sobolev spaces of finite order are also investigated.

In the second part of the work, a numerical method for obtaining approximate solutions is constructed. The convergence is proved, provided that a smooth solution exists. An efficient numerical scheme is proposed.

This work was supported by the Russian Foundation for Basic Research (grants 04-05-64784 and 04-01-00256) and by the Program for Fundamental Research of Presidium of RAS "Mathematical Methods in Nonlinear Dynamics."