  We present a general purpose method for solving partial differential
equations on a closed surface, based on a technique for discretizing the
surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys. 252
(2013), pp. 606-624] which uses projections on coordinate planes. Assuming it
is given as a level set, the surface is represented by a set of points at which
it intersects the intervals between grid points in a three-dimensional grid.
They are designated as primary or secondary. Discrete functions on the surface
have independent values at primary points, with values at secondary points
determined by an equilibration process. Each primary point and its neighbors
have projections to regular grid points in a coordinate plane where the
equilibration is done and finite differences are computed. The solution of a
p.d.e. can be reduced to standard methods on Cartesian grids in the coordinate
planes, with the equilibration allowing seamless transition from one system to
another. We observe second order accuracy in examples with a variety of
equations, including surface diffusion determined by the Laplace-Beltrami
operator and the shallow water equations on a sphere.
