2026 , Volume 31, № 5, p.5-37

Purpose. Accounting for three-dimensional effects leads to more accurate estimation of wave forces on marine structures. The purpose of the present work, which consists of two parts, is a numerical study of the influence of diffraction effects on the main characteristics of interaction between long surface waves such as tsunami waves and large partially immersed structures.

Methods. In the first part of the work, a mathematical formulation is developed within the framework of a 3D potential flow model with a free boundary. The problem of long surface wave interaction with partially immersed or gravity-based structures is formulated in a curvilinear moving coordinate system. A new finite-difference scheme on movable grids is described in detail. The grids adapt to the lateral faces of the fixed structure, to the curvilinear surfaces of its bottom and the basin bed, and to the moving water surface. The difference equations for the velocity potential are derived using a three-dimensional analogue of the integro-interpolation method.

Results. Original second-order approximation formulas for the Neumann condition on impermeable surfaces are derived. These formulas are based on the application of the conservative form of the elliptic equation with variable coefficients for the velocity potential and use its values only at boundary nodes and nodes of the first internal adjacent layer. It is proved that with such approximations, the difference operator for the potential is self-adjoint and inherits this property of the differential operator.

Conclusions. The properties of the finite-difference scheme, established theoretically, ensure that its application will not lead to known numerical effects that distort the values of the computed quantities. The practical significance of the obtained theoretical results will be demonstrated in the second part of this work. Therein, by comparing the results with calculations based on the two- dimensional potential flow model, the importance of accounting for the diffraction of long waves impacting large partially immersed structures will be shown.