A Solution to the Pole Problem for the Shallow Water Equations on a Sphere
Journal article, 2014
We consider a reduced gridding technique for the shallow water
equations on a sphere, based on spherical coordinates. In a small vicinity
of the poles, a longitudinal derivative is discretized at a grid-point
on a parallel, by using points on the great circle through the grid-point
and tangent to the parallel. Centered one-dimensional interpolation
formulas are used in this process and also in connecting adjacent
segments in the reduced grid. The remaining spatial discretization
is obtained by simply replacing derivatives by centered equidistant
finite difference approximations.
Numerical experiments for scalar advection equations and for the
well-known Rossby-Haurwitz test example indicate that the
methods developed work surprisingly well. Some advantages
are that (i) a fairly uniform grid, with many reductions or segments,
can be used, (ii) order of approximation $2p$ in the spatial discretizations
requires only $4p+1$ points and (iii) the local and simple structure
of the schemes will make efficient implementation on massively
parallel computer systems possible.
The paper is an attempt towards global numerical weather
prediction models, by first analyzing the pole problem for reduced
numerical weather prediction.
shallow water equations