The Direct Collocation Meshless Method Based on a Second-Order Phonon Boltzmann Equation for Ballistic–diffusive Phonon Heat Transport

A second-order phonon Boltzmann equation (SOPBE) is proposed based on the phonon Boltzmann equation (PBE) under gray relaxation-time approximation, and the direct collocation meshless (DCM) method is employed to solve the SOPBE. Several numerical tests for phonon transport over a broader range of Knudsen numbers under different boundary conditions are carried out. The results show that the SOPBE solved by the DCM method is applicable for different transport regimes. Moreover, it overcomes the numerical error ray effects of other numerical methods under the ballistic limit to a certain extent. When modeling phonon transport in materials with inhomogeneous acoustic property, the superiority of SOPBE will be more obvious compared with the PBE. The results demonstrate the capability of our methodology for ballistic-diffusive phonon heat transport.