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In the present work a novel approach has been developed for the resolution of this problem of analysis of the movement of roll of a ship. The methodology used for this is based on the modification of the differential equations of the fluid dynamics (RANSE equations), including the movement of the free surface, by applying the finite calculus method. The modified equations are solved using an implicit predictor-corrector scheme and the finite element method (FEM). This resolution scheme is considered optimal for these types of problems, both in accuracy and in calculation time. As an example, it should be noted that for typical problems (more than 1,000,000 finite elements) less than 4 hours of CPU are required to solve several cycles of ship movement (Pentium IV). This allows the analysis of an array of tests in a few days or even hours, having a computer network or sufficient computing power.

An extension from 2D to 3D of the fully-nonlinear numerical wave tank (NWT) used by Hanssen et al. is presented. Potential-flow theory is assumed, and the high-order harmonic polynomial cell (HPC) field method proposed by Shao and Faltinsen is used to solve the governing Laplace equation for the velocity potential ? numerically. The present work deals with an immersed boundary method without and with the use of an octree technique for local grid refinement near the air-water interface, and examines only wave propagation; dealing with wave-body interactions is planned as a future development.