Latent heat thermal energy storage (LHTES) has many applications in engineering fields such as electronic cooling, thermal storage of solar energy, heating and cooling in buildings, waste heat utilization and so on. The advantages of LHTES over sensible thermal energy storage or chemical energy storage techniques are high energy density and phase change at nearly constant temperature. Unfortunately, the low thermal conductivity of PCMs increases the thermal gradient in the energy storage system and impedes the heat transfer efficiency. However, high thermal conductivity fins could be used to promote the melting process in PCM enclosures. As a powerful numerical method developed during the past two decades, lattice Boltzmann method (LBM) was used to simulate the conjugate heat transfer in the solid walls, fins and PCM region. By changing the velocity field and diffusivities, only one distribution function was needed to simulate the melting with natural convection in PCMs and conduction in fins and enclosure surfaces. As a result, the thermal boundary conditions on the interfaces of PCMs, fins and solid walls were satisfied automatically. By using enthalpy-based multiple-relaxation-time (MRT) LBM model, the iteration steps for the latent-heat source term were avoided. Under this case, the conjugate convective heat transfer with phase change is modeled efficiently. The graphics processing units (GPU) computing becomes attractive since the advent of CUDA which includes both hardware and programming environment in 2007. Consequently, the developed MRT LBM code is further implemented to run on GPU. High computation speed was achieved. The melting process in PCMs was investigated for different materials of fins and walls, number of fins, fin configurations, hot wall temperature, thermal boundary conditions, and inclination angle of the PCM cavity. Lattice Boltzmann method implemented on GPU was demonstrated as an efficient approach to study the PCM melting process with internal fins. (C) 2016 Elsevier Ltd. All rights reserved.