The Cartesian method for solving partial
differential equations in spherical geometry
by P. N. Swarztrauber, D. L. Williamson, and J. B. Drake,
Dynamics of Atmospheres and Oceans, 27
(1997), pp. 679-706.
Cartesian coordinates are used to solve the nonlinear shallow-water
equations on the sphere. The two-dimensional equations, in spherical
coordinates, are first embedded in a three-dimensional system in a
manner that preserves solutions of the two-dimensional
system. Solutions of the three-dimensional system, with appropriate
initial conditions, also solve the two-dimensional system on the
surface of the sphere. The higher dimensional system is then
transformed to Cartesian coordinates. Computations are limited to the
surface of the sphere by projecting the equations, gradients, and
solution onto the surface. The projected gradients are approximated
by a weighted sum of function values on a neighboring stencil. The
weights are determined by collocation using the spherical harmonics in
trivariate polynomial form. That is, the weights are computed from the
requirement that the projected gradients are near exact for a small
set of spherical harmonics. The method is applicable to any
distribution of points and two test cases are implemented on an
icosahedral geodesic grid. The method is both vectorizable and
Last updated July 14, 1998.
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