ISSN 2542–0380 
Труды Института механики им. Р.Р. Мавлютова
Электронный научный журнал  Electronic Scientific Journal
Proceedings of the Mavlyutov Institute of Mechanics

The efficiency of the secondorder accurate UNO and TVDmodifications of the Godunov method is compared using a number of problems on the propagation of linear waves in an elastic body, their interaction with each other and with the surface of the body. In particular, onedimensional problems having analytic solutions and a twodimensional problem of the dynamics of a body in the vicinity of the impact domain on its free surface are considered. It is shown that if in the problems there are wellmarked extrema or short waves, then the UNOscheme is more effective, since in such cases decrease in the accuracy of the TVDscheme becomes apparent due to strictly satisfying the TVD condition. Because of approximately satisfying the TVD condition, the UNOscheme can lead to the appearance of oscillations of numerical nature at the level of approximation errors. However, this does not reduce the efficiency of the UNO scheme since the amplitude of those oscillations decreases with refinement of the grid.
linear waves in an elastic body,
S.K. Godunov method,
UNOschemes,
TVDschemes
Purpose. To compare the efficiency of computing linear waves in an elastic body employing the secondorder accurate UNO and TVD modifications of the Godunov method.
Methodology. To estimate the efficiency of the modifications under consideration, we use solutions to onedimensional problems, having an analytic solution, on the propagation of waves in the body, their interaction with each other and with the surface of the body as well as the twodimensional problem of the dynamics of the body under impact on its free surface. The obtained numerical solutions are compared with each other in the areas of their monotonous change, extrema and discontinuities at different instants of time.
Findings. It is found that if in the problem solutions there are wellmarked extrema or short waves, then the UNOscheme is more effective due to the fact that in such cases the accuracy of the TVDscheme decreases from the second order to the first order. It is shown that when using the UNOscheme, the appearance of oscillations of numerical nature at the level of approximation errors is possible, which does not affect the efficiency of this scheme since the amplitude of those oscillations decreases with refinement of the grid.
Originality/value. When computing linear waves in an elastic body in the presence of wellmarked extrema or short waves in the solution, it is preferable to use the UNOmodification of the Godunov method since TVDschemes in such problems can