On numerical solution of Poincaré problem for oceanic circulations (Q2365846)

A numerical method for the solution of an initial value problem of ocean dynamics based on splitting the problem's main operator into a sequence of simpler ones is discussed. The ocean is represented by a cylinder of constant depth. The governing differential equations containing the velocity components and pressure and density deviations from standard values are introduced in a linear version. The boundary conditions at the ocean surface, at the shore and at the ocean bottom as well as the initial data are established. The solution of the problem under consideration is found using the method of orthogonal expansions, at which the components of the system's solution are evaluated in the form of a Fourier series. This leads to a spectral problem with a certain basis of eigenfunctions corresponding to a system of positive eigenvalues. Finally, a set of problems for the Fourier coefficients is arrived, which do not depend on the ocean depth coordinate anymore. To get a numerical algorithm for the solution of these problems, approximations are introduced for a certain time interval, which finally together with the appropriate boundary conditions lead to a classical Poincaré problem of Fredholm type. The numerical algorithm is formulated by representing the governing equation in a finite-difference form. The way of building this algorithm is described in detail. It results in a system of linear algebraic equations, the coefficient matrix of which can be splitted into a sum of two matrices, each of them representing a more simple type of equations. Finally, the iteration method for the equation with the two separated matrices is introduced. It is shown that the iteration process converges to the exact solution of the equation system with the coefficients matrix not being splitted.