Semi-implicit Lax-Wendroff kinetic scheme for multi-scale phonon transport

Fast and accurate predictions of the spatiotemporal distributions of temperature are crucial to the multi-scale thermal management and safe operation of microelectronic devices. To realize it, an efficient semi-implicit Lax-Wendroff kinetic scheme is developed for numerically solving the transient phonon Boltzmann transport equation (BTE) from the ballistic to diffusive regime. The biggest innovation of the present scheme is that the finite difference method is used to solve the phonon BTE for the reconstruction of the interfacial distribution function at the half-time step, where the second-order numerical schemes are used for both the temporal and spatial discretization. Consequently, the phonon scattering and migration are coupled together within one time step, and the evolution process of phonon distribution function follows the actual physical law even if the time step is much longer than the relaxation time. Numerical results show that the present scheme could accurately predict the steady/unsteady heat conduction in solid materials from the ballistic to diffusive regime, and its time step or cell size is not limited by the relaxation time or phonon mean free path. The present work could provide a useful tool for the efficient predictions of the macroscopic spatiotemporal distributions in the multi-scale thermal engineering.