Finite Element Method for Internal Wave Equation for Stratified Fluid

Dawletbay Utebaev


In many areas of natural science non-stationary problems on internal wave motion arise. For example, such problems appear in aerophysics, geophysics, oceanology, in theory of rotating fluid, and at design and construction of mining constructions (Gabov & Sveshnikov, 1990). It is a complicated mathematical problem to obtain accurate analytical solutions for such problems. In such cases a natural apparatus for study of internal wave motion processes is in numerical methods. In this work, the schemes of the finite element method with high accuracy in space and time for solution of a mixed boundary problem for internal wave equation for stratified fluid are proposed and studied. The schemes constructed have specific advantages compared to the other schemes: a scheme with high order of accuracy (more than two); besides the solution itself one finds along which that its derivative (velocity) with the same accuracy; at use of interpolation representation of the solution one can obtain, if needed, the solution and its derivative for an arbitrary instant; since the schemes are two-layer ones, one can use variable step without loss of accuracy; the scheme is conventionally stable and requires 4 times more arithmetical operations compared to ordinary ones, though this scheme makes it possible to choose larger time steps to get given accuracy. Besides that, evaluations of accuracy of the schemes for the problem under consideration are obtained. By means of dispersion analysis comparison to known schemes is carried out.

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