The saturation of stimulated Brillouin scattering (SBS) by the decay to turbulence of the ion acoustic wave (IAW) that participates in the three-wave SBS interaction is demonstrated using a quasinoiseless one-dimensional numerical solution to the Vlasov-Maxwell system of equations. This simulation technique permits careful examination of the decay process and its role in the complex evolution of SBS. The IAW decay process is shown to be an effective SBS saturation mechanism. In our example, the instantaneous plasma reflectivity saturates at similar to 30% and drops to similar to 0% as a direct consequence of IAW decay. A contrasting example where the reflectivity is controlled by dephasing due to the nonlinear frequency of the IAWis also discussed. (C) 2015 AIP Publishing LLC.