We present a new spectral scheme for analysing functions of half-integer spin-weight on the $2$-sphere and demonstrate the stability and convergence properties of our implementation. The dynamical evolution of the Dirac equation on a manifold with sp
atial topology of $mathbb{S}^2$ via pseudo-spectral method is also demonstrated.
The purpose of this note is to point out that a naive application of symplectic integration schemes for Hamiltonian systems with constraints such as SHAKE or RATTLE which preserve holonomic constraints encounters difficulties when applied to the nume
rical treatment of the equations of general relativity.
