The convergence to the steady state of an implicit unstructured density-based compressible Navier-Stokes flow solver shows many cases exhibiting stalling or slower convergence in the later part mainly due to fluctuation in the local error. This causes a delay in convergence and often prevents machine zero convergence from being achieved. This paper presents the use of local error control for acceleration of later convergence. This method is tested against standard benchmark problems, and then on a more complex case of flow over an airfoil taken from recent published work. The results show that the method is capable of very rapid acceleration to machine zero convergence. The acceleration has been found to be of an order of 500-1000% faster in comparison to the same implicit scheme with Courant-Friedrichs-Lewy-based time stepping and no local error control. The present method acts on top of the implicit solver, and thus its implementation requires very little modification to an existing code. © 2018 by the American Institute of Aeronautics and Astronautics, Inc.