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We present the first numerical code based on the Galerkin and Collocationmethods to integrate the field equations of the Bondi problem. The Galerkinmethod like all spectral methods provide high accuracy with moderatecomputational effort. Several numerical tests were performed to verify theissues of convergence, stability and accuracy with promising results. This codeopens up several possibilities of applications in more general scenarios forstudying the evolution of spacetimes with gravitational waves.

Numerical Evolution of axisymmetric vacuum spacetimes: a code based on the Galerkin method