Computing with Functions in Spherical and Polar Geometries

Spherical and polar geometries are ubiquitous in computational science and engineering, arising in, for example,

weather and climate forecasting, geophysics, and astrophysics. Central to many of these applications is the task

of developing efficient and accurate approximations of functions defined on the surface of the unit sphere or

on the disk. We present a new low rank method for this task by combining an iterative, structure-preserving

variant of Gaussian elimination together with the classic double Fourier sphere method. The resulting scheme

gives a compressed representation functions on the sphere or disk, ameliorates oversampling issues near the

poles of the sphere or origin of the disk, and converges geometrically for sufficiently analytic functions. The low

rank representation makes operations such as function evaluation, differentiation, and integration particularly

efficient. We illustrate the applicability of our method to common computational tasks from data analysis, vector

calculus, and the solution of partial differential equations, which are all implemented in the new Spherefun and

Diskfun features of the Chebfun software system (www.chebfun.org).