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

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

inserting 20 into 19 and then dividing by yields:

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:

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

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 |

2010-05-19