About solvability and numerical simulation of
nonstationary flow of incompressible fluid with
a free surface
P.P. Shirshov Institute of Oceanology
of the Russian Academy of Sciences,
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."