Semi-implicit Lax-Wendroff kinetic scheme for hydrodynamic phonon transport

A semi-implicit Lax-Wendroff kinetic scheme is developed for hydrodynamic phonon transport in solid materials based on the Boltzmann transport equation under the double relaxation time approximation, in which both the normal and resistive scattering processes are accounted. The trapezoidal and midpoint rules are adopted for the temporal integration of the scattering and migration terms under the framework of finite volume method, respectively. Instead of direct numerical interpolation, the kinetic equation is solved again when reconstructing the interfacial flux, in order to realize the coupling of phonon migration and scattering within a numerical time step. Specifically, the finite difference scheme is introduced and the second-order upwind or central schemes are used for the reconstruction of the interfacial distribution function and its spatial gradient. Consequently, the cell size and time step of the present method could be larger than the phonon mean free path and relaxation time in the limit of small Knudsen numbers. Numerical tests demonstrate that the present method can accurately capture multi-scale thermal conduction phenomena within different normal or resistive scattering rates.