## Accuracy of finite differences in smooth media

**Ludek Klimes**
### Summary

Accuracy of finite-difference schemes of the
2nd and 4th order in 2-D and 3-D regular
rectangular grids is studied. The method of designing the
schemes and estimating their accuracy is proposed.
Considered are the point schemes, expressed in terms of
the values of the material parameters and of the wavefield at
gridpoints. Only the common schemes applicable in
smooth parts of seismic models, outside structural interfaces
are taken into account.
Finite differences at structural interfaces are studied in another paper.

The inaccuracy of finite-difference schemes is governed, above all,
by the error in the propagation velocity, caused by the discretization.
This error is estimated for several finite-difference schemes.
It is explicitly
dependent on the direction of propagation and on the wave polarization.
The maximum propagation-velocity error over all directions of propagation
enables to appreciate the accuracy of individual schemes in order
to find the best one. The proposed approach is general, and applicable
to other finite-difference schemes, for example, of the
6th and higher orders.

### Keywords

Seismic waves, finite differences of the 2nd and 4th order,
2-D and 3-D seismic modelling, elasticity.

### Whole paper

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### Revised version

Klimes, L.:
Accuracy of finite differences in smooth media.
PAGEOPH, **148** (1996), 39-76.

In: Seismic Waves in Complex 3-D Structures, Report 1,
pp. 127-149, Dep. Geophys., Charles Univ., Prague, 1994.

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