Numerical simulation of elastic wave propagation using a finite volume method

Emmanuel Dormy and Albert Tarantola

Département de Sismologie, Institut de Physique du Globe de Paris, France

Journal of Geophysical Research, Vol. 100, No. B2, pp. 2123-2133.

Abstract. Like the finite difference method, the finite volume method gives an approximate value for the derivative of a field at a given point using the values of the field at a few locations neighboring the point. The method uses the divergence theorem, considers a ``finite volume'' around the point and discretizes the surface bounding the volume. When the considered finite volumes are regular polyhedra, one obtains the expressions corresponding to standard centered finite differences, but the finite volume method is more general than the finite difference method because it may deal directly with irregular grids. It is possible to give a finite volume formulation of the elastodynamic problem, using dual volumes, that correspond, in the regular case, to the staggered grids used in the finite difference method. The scheme thus obtained is more general than the one obtained using finite differences, as the ``grids'' may be totally unstructured, but at the cost of having, in the general case, only a first order accuracy. Although the scheme is not consistent, numerical tests suggest that it is stable and convergent. This implementation of a finite volume method does not provide a way for a more general treatment of the boundaries than the conventional finite difference method.

Color version of figures 11 and 12:

Figure 11. Numerical simulation of the propagation of elastic waves, using the finite volume method in a Cartesian grid. In the left hand side, the grid is regular and, then, the method is equivalent to the finite difference method. In the right hand side the grid has been randomly perturbed except a rank of cells on the border. The top of the figure shows a detail of the grids (the full grid has 256x256 points). The middle and bottom of the figure show two snapshots of the simulation at two different time steps(the boundary conditions being periodical, two periods are shown per each dimension). The variable represented is the trace of the strain tensor epsilon_ii. The color code used has been chosen to emphasize small values, in order to clearly display the numerical artifacts. It is interesting to note that, even for strongly perturbed grids, the numerical artifacts are surprisingly small.

Figure 12. Numerical simulation of the propagation of elastic waves, using the finite volume method in a minimal grid. In the left hand side, the grid is regular and, then, the method is equivalent to the finite difference method in a minimal grid. In the right hand side the grid has been randomly perturbed except a rank of cells on the border. The top of the figure shows a detail of the grids (the full grid has 512x512 points). The middle and bottom of the figure show two snapshots of the simulation at two different time steps. The top is a free surface (Neumann type), while the bottom and the sides are rigid (Dirichlet type). The variable represented is the trace of the strain tensor epsilon_ii. The color code used has been chosen to emphasize small values, in order to clearly display the numerical artifacts. Even for strongly perturbed grids, the numerical artifacts are surprisingly small.

An animated rhombododecahedron (see Fig. 7 of the article).