Computation of two-dimensional disturbances in an elastic body

A. A. Aganin   ; N. A. Khismatullina

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 2, pp. 143-160

Résumé

The classical Godunov method is widely used for the numerical study of waves in continuous media. If the Courant condition is satisfied, the Godunov scheme is stable and monotonous. However, due to its first order of accuracy it can lead to rather large smearing of jumps, contact discontinuities, and other features of the solution in the domains where the solution gradients are great. In this paper, the possibility of increasing the efficiency of computation of linear waves in an elastic body by applying one of the second-order accurate UNO modifications of the classical Godunov method has been studied. In that UNO modification, the monotonicity requirement is replaced by the TVD condition, and all the parameters inside each grid cell are assumed linear rather than constant as they are in the Godunov method. The TVD condition is met just on the level of approximation. To derive the second order of accuracy in time, the time derivatives of the unknown functions have been expressed in terms of their spatial derivatives. Those expressions allow to calculate the next half-time-layer values of the unknown functions at the center of the grid cells and on both sides of the cell boundaries. The values of the unknown functions on the boundaries themselves have been found by solving the corresponding Riemann problems. Subsequently, numerical flows across the cell boundaries have been computed. Computation of the values at the next time layer has been carried out by an explicit scheme. Some limiters for the values of the spatial derivatives have been introduced. The present UNO scheme limiters use approximations to both the first and second derivatives. The efficiency of the proposed UNO modification has been estimated by computing a number of one- and two-dimensional problems on propagation of linear waves in an elastic body and their interaction with each other and with the surface of the body. The results of those computations have been compared with the exact solutions and the results of applying the Godunov method. It has been shown that the UNO scheme considered allows one to reduce computational costs by more than a factor of ten.

Détail

Keywords: UNO scheme, Godunov scheme, efficiency of difference schemes, linear wave, elastic body.

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