On the finite difference solution of two-dimensional induction problems

Abstract

The numerical solution by finite differences of two-dimensional problems in electromagnetic induction is reexamined with a view to generalizing the method to three-dimensional models. Previously published work, in which fictitious values were used to derive the finite difference equations, is discussed and some errors in the theory which appear to have gone undetected so far, are pointed out. It is shown that the previously published Bpolarization formulas are incorrect at points where regions of different conductivity meet, and that the E-polarization formulas are inaccurate when the step sizes of the numerical grid around the point are uneven. An appropriately-modified version of the two-dimensional theory is developed on the assumption that the Earth's conductivity is a smoothly-varying function of position, a method which naturally lends itself to three-dimensional generalization. All the required finite-difference formulas are derived in detail, and presented in a form which is suitable for programming. A simple numerical calculation is given to illustrate the application of the method and the results are compared with those obtained from previous work.

Two-dimensional modelling of resistivity structure beneath the Tohoku district, northern Honshu of Japan, by a finite element method.

“…In the problem of electromagnetic induction, the finite difference method is widely used (see, for example, JONES and PASOOE, 1971;BREWITT-TAYLOR and WEAVER, 1976), Since this method is based on rectangular meshes, a problem with complicated boundaries is not easily solved. In order to estimate the effect of the ocean, which is very thin compared with low resistivity material in the crust and the mantle, the mesh size must be taken to be particularly small.…”

The frequency response of two-dimensional induction anomalies revisited.

“…The finite difference method which we used was described by BREWITT-TAYLOR and WEAVER (1976) and the integral equation method was described by BERDICHEVSKY and ZHDANOV (1984). In the integral equation method, if the time…”

Two-Dimensional Magnetotelluric Modelling of Ore Deposits: Improvements in Model Constraints by Inclusion of Borehole Measurements

“…To obtain correct finitedifference results for the lower boundary of the computational domain that coincides with the top of a perfectly conductive half space as in the control model, Neumann boundary conditions need to be used at the lower edge of the mesh in the TM-mode (Brewitt-Taylor and Weaver 1976; Weaver et al 1985). Brewitt-Taylor and Weaver (1976) implemented Neumann boundary conditions using single sided approximations, reducing the order of the finite-difference approximation to first order. To achieve better computational accuracy, the approach presented here follows LeVeque (2007) and uses a centred second-order approximation of the first derivative employing ghost nodes one fictitious cell height below the lower boundary of the computational domain.…”