FINITE ELEMENT METHOD SCHEMES OF HIGHER ACCURACY FOR SOLVING NON-STATIONARY FOURTH-ORDER EQUATIONS (ҚазҰУ Хабаршысы. Математика, механика, информатика сериясы, 2023)
High-order Sobolev-type equations are mathematical models used in many applied problems. As is known, in many cases, it is difficult to obtain analytical solutions to high-order equations; therefore, they are mainly solved by numerical methods. At present, the method of straight lines is often used to solve non-stationary problems of mathematical physics; in this method, discretization is first realized only in spatial variables, and the resulting system of ordinary differential equations of high dimension is solved by finite difference methods or finite elements of higher accuracy. In this study, for a system of ordinary differential equations of the fourth order, new multi-parameter difference schemes of higher accuracy based on the finite element method are constructed. The presence of parameters in the scheme makes it possible to regularize the schemes in order to optimize the implementation algorithm and the accuracy of the scheme. The stability and convergence of the constructed difference schemes are also proved, and accuracy estimates are obtained on their basis. An algorithm for the implementation of the constructed difference schemes is presented. The results obtained can be further applied in the numerical solution to initial-boundary value problems for the equations of dynamics of a compressible stratified rotating fluid, magnetic gas dynamics, ion-acoustic waves in a magnetized plasma, spin waves in magnets, cold plasma in an external magnetic field, etc.