Труды Института механики им. Р.Р. Мавлютова

Aganin A.A., Khismatullina N.A. Schemes of the second order accuracy for computing the dynamics of disturbances in an elastic body. Proceedings of the Mavlyutov Institute of Mechanics. 2017. 12(1). 44–50.

2017. Vol. 12. Issue 1, Pp. 44–50
URL: http://proc.uimech.org/uim2017.1.007,en
DOI: 10.21662/uim2017.1.007

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

Aganin A.A., Khismatullina N.A.

Institute of Mechanics and Engineering, Kazan

Abstract

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.

Keywords

linear waves in an elastic body,
S.K. Godunov method,
UNO-schemes,
TVD-schemes

Article outline

Purpose. To compare the efficiency of computing linear waves in an elastic body employing the second-order accurate UNO- and TVD- modifications of the Godunov method.

Methodology. To estimate the efficiency of the modifications under consideration, we use solutions to one-dimensional 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 two-dimensional 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 well-marked extrema or short waves, then the UNO-scheme is more effective due to the fact that in such cases the accuracy of the TVD-scheme decreases from the second order to the first order. It is shown that when using the UNO-scheme, 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 well-marked extrema or short waves in the solution, it is preferable to use the UNO-modification of the Godunov method since TVD-schemes in such problems can