## Heterogeneous formulations of
elastodynamic equations and finite-difference schemes

**Jiri Zahradnik** **&**
**Enrico Priolo**
### Summary

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. Some of the
heterogeneous schemes are justified for a given problem, but not
all, in general; 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. The 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.

### Revised version

Zahradnik, J. & Priolo, E.:
Heterogeneous formulations of
elastodynamic equations and finite-difference schemes.
Geophys. J. Int., **120** (1995), 663-676.

In: Seismic Waves in Complex 3-D Structures, Report 1,
pp. 187-214, Dep. Geophys., Charles Univ., Prague, 1994.

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