A numerical algorithm for acoustic noise predictions based on solving Lilley's third order wave equation in the time-space domain is developed for a subsonic axisymmetric jet. The sound field is simulated simultaneously with the source field calculation, which is based on a direct solution of the compressible Navier-Stokes equations. The computational domain includes both the nearfield and a portion of the acoustic farfield. In the simulation, the detailed sound source structure is provided by the nearfield direct numerical simulation (DNS), while the sound field is obtained from both the DNS and the numerical solution to the non-linear Lilley's equation. The source terms of Lilley's equation are used to identify the apparent sound source locations in the idealized axisymmetric low-Reynolds number jet. The sound field is mainly discussed in terms of instantaneous pressure fluctuations, frequency spectra, acoustic intensity and directivity. A good agreement is found between the predictions from the axisymmetric Lilley's equation and the DNS results for the sound field. Limitations and perspectives of the simulation are also discussed.