Finite Element Solution of a Problem for Gravity Gyroscopic Equation in the Time Domain

Abstract

To solve the equation for gravity-gyroscopic waves in a rectangular domain, the distinguished algorithm for the solution of the Cauchy problem for a second-order transient equation is proposed. This algorithm is developed by using the time-varying finite element method. The space derivatives in the gravity-gyroscopic wave equation are approximated with finite differences. The stability and accuracy of the method are estimated. The procedure for the implementation of the method is developed. The calculations were performed for determining the steady-state modes of fluctuations of the solutions of the gravity-gyroscopic wave equation depending on the problem parameters.

Solution of the Neumann Problem with Respect to the Equation for Gravity-Gyroscopic Waves by the Finite Element Method

“…The first conclusion coincides with the conclusion in our paper [6], that is, there is no decay of oscillations according to solution ox (see Figure 19).…”

“…The differences of the approximate solution of the problem (1-3) from the solution in our paper [6] are the following:…”

“…b) The approximation with respect to spatial variables (in [6] a net method is used). As a result, the template for the internal net points consists of 9 nodes instead of 5 nodes in [6]. c) Matrix operators , D A are degenerate operators, that is, zero eigenvalues 0…”

“…The approximate solution of problem (6) is assumed to be expressed as a third degree Hermite spline, as in [6]:…”

“…In terms of hydrodynamics, Neumann boundary conditions (2) correspond to no-flow of liquid through boundary  . In our paper [6], Dirichlet conditions 0, u x  , which correspond to free flow of liquid through the boundary, were assumed [1]. Let us determine the general solution of problem (1-3) as function…”