Schemes of the second order accuracy for computing the dynamics of disturbances in an elastic body

The efficiency of the second-order accurate UNO- and TVD-modifications 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, one-dimensional problems having analytic solutions and a two-dimensional 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 well-marked extrema or short waves, then the UNO-scheme is more effective, since in such cases decrease in the accuracy of the TVD-scheme becomes apparent due to strictly satisfying the TVD condition. Because of approximately satisfying the TVD condition, the UNO-scheme 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.