Implicit finite difference in time-space domain with the helix transform

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## Stability analysis

The Von-Neumann stability analysis is useful in predicting the largest time steps possible for a particular order of a finite-difference scheme, for which the wavefield will not diverge. The application of this analysis to the 2nd order in time and space explicit finite-difference approximation of the 2-way wave equation follows. Assuming , and using as a time index, the approximation is:

 (19)

The field being propagated is some function of time combined with a harmonic function of space:

 (20)

inserting 20 into 19 and then dividing by yields:

 (21)

where

In order to have stable propagation, the amplification factor - the amplitude ratio between the future wavefield and the current wavefield, must be smaller or equal to 1. This is also a requirement for the ratio between the past wavefield and the current wavefield, as the time reversed wavefield must also remain stable. From this consideration we have:

 (22)

Dividing (21) by , and using the trigonometric identity for cosine we get:

 (23)

is bounded by , and the requirement is that the amplification factor . It follows that

 (24)

Since , this analysis provides us with a way to determine the maximum time step for a given minimum velocity and spatial differencing step.
The same derivation for the 2D implicit finite differencing weights as derived in Equation 17 yields the amplification factor:

 (25)

The boundaries for remain . Because is necessarily positive, it follows that for any in this 2D implicit finite-difference scheme. The time step can be arbitrarily large without causing the wavefield to diverge. This, of course, does not mean that we will get a wavefield with any arbitrary .

 Implicit finite difference in time-space domain with the helix transform

Next: Conclusion and future work Up: Standard implicit propagation Vs. Previous: Non-separability of velocity from

2010-05-19