This thesis presents work towards a global numerical evolution of the linearised conformal field equations near spatial infinity. We describe the mathematical framework for the initial boundary problem formulation of the conformal field equations. Numerical methods are developed to avoid the singular behaviour of the equations at the critical sets at the top and bottom of the cylinder at spatial infinity.

Implicit numerical methods, along with the application of adaptive step sizes, are successfully applied to evolve initial data from past null infinity on to the cylinder. Then, a combination of spatial and temporal evolution equations are used to evolve around the future critical set and extract the data on future null infinity. We also consider briefly the application of these techniques to a fully compactified setting.