The IPv6 Routing Protocol for Low-power and Lossy Networks (RPL) has been recently developed by the Internet Engineering Task Force (IETF). Given its crucial role in enabling the Internet of Things, a significant amount of research effort has already been devoted to RPL. However, the RPL network convergence process has not yet been investigated in detail. In this paper we study the influence of the main RPL parameters and mechanisms on the network convergence process of this protocol in IEEE 802.15.4 multihop networks. We also propose and evaluate a mechanism that leverages an option available in RPL for accelerating the network convergence process. We carry out extensive simulations for a wide range of conditions, considering different network scenarios in terms of size and density. Results show that network convergence performance depends dramatically on the use and adequate configuration of key RPL parameters and mechanisms. The findings and contributions of this work provide a RPL configuration guideline for network convergence performance tuning, as well as a characterization of the related performance trade-offs.