In theory this equation could also be solved by a spectral method on S2, using spectral differentiation to evaluate the angular derivatives. (See [70, 75, 37Jump To The Next Citation Point, 12, 11, 10, 13, 35, 33, 34, 77, 95, 96, 97, 120, 124, 123, 122] for further discussion of spectral methods.) This should yield a highly efficient apparent horizon finder. However, I know of no published work taking this approach.