On peculiarities of the Lebedev scheme for simulation of elastic wave propagation in anisotropic media

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Main attention is being given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the scheme approximates a system of equations which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts”. Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of a problem on a sufficiently smooth solution. The derived results are of importance for the construction of reflectionless boundary conditions, development of a heterogeneous Lebedev scheme, approximation of point sources, etc.