Refined finite‐difference simulations using local integral forms: Application to telluric fields in two dimensions

Abstract

This paper describes a new finite‐difference form for simulating the behavior of telluric fields near electrical inhomogeneities. The technique involves a local integration of the electric current density crossing a closed surface surrounding a mesh node. To illustrate the concept, a two‐dimensional (2-D) model is considered, but it is readily possible to generalize to three dimensions. The resulting expressions, which are accurate to second degree everywhere, have the form of nine‐point finite‐difference operators, but they have a higher precision than those derived from the usual differential forms which result in five‐point operators. In particular, the new form accounts for cross‐derivative [Formula: see text] effects in the region about each node. Including this term can provide significant improvements in accuracy near sharp, localized discontinuities, where the anomalous field decays rapidly (as 1/r or [Formula: see text]) with distance. An analytical solution is compared to finite‐difference calculations using both the conventional five‐point differential form and the new nine‐point integral form developed here. The results suggest that, in some cases, one might expect at least a factor of three improvement when using the nine‐point operator instead of the five‐point operator. This is particularly true in the vicinity of localized structures where the curvilinear character of the distorted field is most pronounced and one would expect the cross‐derivative term to be large.