On Certain Truncation Errors Associated with Spherical Coordinates

Abstract

Excessive errors are committed in grids, in which mesh-lengths in the longitudinal direction are preserved in polar regions, if the hydrodynamic equations written in spherical coordinates are directly transformed into finite differences. The errors arise from the curvilinearity of the coordinate system. One alternative is the abandonment of spherical coordinates, with the retention of the constant mesh-length grid, an alternative which is not economical. The errors can be reduced by at least two orders of magnitude by adoption of a grid regular in latitude and longitude angle, with the consequent great space resolution in polar regions. Even the latter alternative may result in unacceptable truncation errors in finite-difference equations in spherical co-ordinates. It turns out, however, that if one departs from spherical coordinates to the extent of expressing velocity components in Cartesian coordinates on locally tangent planes, a further reduction of error by an order of magnitude is achieved, a reduction to perhaps acceptable levels.