Heterogeneous formulations of elastodynamic equations and finite-difference schemes

The heterogeneous formulation of differential equations is justified in this paper. This means that the material discontinuities, properly introduced into the elastodynamic equations of motion, manifest themselves as (delta function) localized body forces, serving for the traction continuity. The traction-continuity conditions, formulated separately from the differential equations of motion, are not needed. This result is independent of the method used to solve the differential equations, and encourages attempts to construct heterogeneous formulations of finite-difference equations. A particular heterogeneous finite-difference scheme can be justified by the Taylor expansion method. In general some of the heterogeneous schemes are justified for a given problem, but not all; some of the heterogeneous schemes even violate the traction-continuity condition. A recent elastic scheme (PS2) has been theoretically justified for the problems characterized by a free surface and/or an interface parallel to grid-line direction, including a discontinuity reaching the surface. Synthetic seismograms computed with the PS2 scheme have been compared with the exact solution and the higher-order spectral-element solution. Attention has been focused on the Rayleigh and interface waves. A good agreement between the compared solutions has been found for all presented test models.