We present a high-order scheme for solving the full non-linear Einstein equations on characteristic null hypersurfaces using the framework established by Bondi and Sachs. This formalism allows asymptotically flat spaces to be represented on a finite, compactified grid, and is thus ideal for far-field studies of gravitational radiation. We have designed an algorithm based on 4th-order radial integration and finite differencing, and a spectral representation of angular components. Consequently the scheme offers more accuracy at a given computational cost compared to previous methods which are second-order accurate. Based on a newly implemented code, we show that the new numerical scheme remains stable and is convergent at the expected order of accuracy.