  Many applications in science call for the numerical simulation of systems on
manifolds with spherical topology. Through use of integer spin weighted
spherical harmonics we present a method which allows for the implementation of
arbitrary tensorial evolution equations. Our method combines two numerical
techniques that were originally developed with different applications in mind.
The first is Huffenberger and Wandelt's spectral decomposition algorithm to
perform the mapping from physical to spectral space. The second is the
application of Luscombe and Luban's method, to convert numerically divergent
linear recursions into stable nonlinear recursions, to the calculation of
reduced Wigner d-functions. We give a detailed discussion of the theory and
numerical implementation of our algorithm. The properties of our method are
investigated by solving the scalar and vectorial advection equation on the
sphere, as well as the 2+1 Maxwell equations on a deformed sphere.
