  We introduce a single patch collocation method in order to compute solutions
of initial value problems of partial differential equations whose spatial
domains are 3-spheres. Besides the main ideas, we discuss issues related to our
implementation and analyze numerical test applications. Our main interest lies
in cosmological solutions of Einstein's field equations. Motivated by this, we
also elaborate on problems of our approach for general tensorial evolution
equations when certain symmetries are assumed. We restrict to U(1)- and Gowdy
symmetry here.
