  We present a finite difference version of the eth formalism, which allows use
of tensor fields in spherical coordinates in a manner which avoids polar
singularities. The method employs two overlapping stereographic coordinate
patches, with interpolations between the patches in the regions of overlap. It
provides a new and effective computational tool for dealing with a wide variety
of systems in which spherical coordinates are natural, such as the generation
of radiation from an isolated source. We test the formalism with the evolution
of waves in three spatial dimensions and the calculation of the curvature
scalar of arbitrarily curved geometries on topologically spherical manifolds.
The formalism is applied to the solution of the Robinson-Trautman equation and
reveals some new features of gravitational waveforms in the nonlinear regime.
