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In this paper we consider the single patch pseudo-spectral scheme fortensorial and spinorial evolution problems on the 2-sphere presented in [3,4]which is based on the spin-weighted spherical harmonics transform. We apply andextend this method to Einstein's equations and certain classes of sphericalcosmological spacetimes. More specifically, we use the hyperbolic reductions ofEinstein's equations obtained in the generalized wave map gauge formalismcombined with Geroch's symmetry reduction, and focus on cosmological spacetimeswith spatial S3-topologies and symmetry groups U(1) or U(1) x U(1). We discussanalytical and numerical issues related to our implementation. We test our codeby reproducing the exact inhomogeneous cosmological solutions of the vacuumEinstein field equations obtained in [7].

Numerical solutions of Einstein's equations for cosmological spacetimes with spatial topology S3 and symmetry group U(1)