▎ 摘　　要

The purpose of this paper is to study thermal effects in a graphene sheet on substrate. The temperature evolution of the phonon branches and of the crystal lattice is obtained when also the presence of the substrate is taken into account. The numerical strategy is based on the coupling of stochastic and deterministic numerical methods; Boltzmann equation for charge transport is treated by using the Direct Simulation Monte Carlo approach, whose results are treated as source terms for the phonon Boltzmann equation which is solved deterministically. The results are useful also for applications as correct design of graphene-based prototypes, when heating effects in the presence of a substrate are relevant.