ISSN:
1365-246X
Source:
Blackwell Publishing Journal Backfiles 1879-2005
Topics:
Geosciences
Notes:
Solving by finite differences for the magnetic field in an E-polarization induction problem is an unorthodox approach because it is simpler to obtain the solution for the single component of the electric field, from which the two magnetic components are readily found by differentiation. Nevertheless, when the E-polarization model appears as the 2-D limit on the boundary of some general 3-D problem in which a finite-difference method is being used to solve for the magnetic field, then the unorthodox approach arises quite naturally. In a previous paper on this subject, it was shown that this problem is beset with numerical difficulties, and a 9-point formula which incorporated the divergence-free nature of the magnetic field was proposed as a way of eliminating numerical instabilities and improving the accuracy of calculation for certain extreme models. For others, however, namely those with very high conductivity contrasts near the surface, it has since been found that even the 9-point formulae are not very accurate. Here we report on further improvements to the finite-difference formulae which lead to accurate results for these types of models while still preserving all the advantages offered by the previous modifications. Two methods are described and compared. In the first, a standard fixed grid is used and new 9-point formulae are derived which give both magnetic field components at each node, and a simpler and more effective finite-difference equation is obtained for one of the surface boundary conditions. In the other approach, a staggered grid is used in which the electric and two magnetic components each have their own grid points. Methods are also developed for calculating the surface electric field from the magnetic field solutions obtained on both types of grid. Calculations arc performed for two different models which represent the types of configurations that have caused numerical problems in the past. It is found that both approaches give excellent results when compared with those generated by standard 2-D finite-difference programs which solve directly for the electric field, and since the numerical problems overcome are similar to those found in a general 3-D induction problem, it is concluded that fixed grids could also be used in three dimensions with an accuracy comparable to that provided by staggered grids.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1111/j.1365-246X.1996.tb05301.x
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