Lattice Boltzmann and Discrete Ordinates Methods for Phonon Transport Modeling: A Comparative Study

The description of heat transport at small length scales is very important in understanding a wide range of micro and nanoscale systems. In systems where coherent phonon transport effects are negligible, the Boltzmann transport equation (BTE) is often employed to describe the distribution and propagation of thermal energy in the lattice. The phonon distribution function depends not only on the temporal and spatial coordinates, but also on polarization and wave vector, making fully-resolved simulations very expensive. Therefore, there is a need to develop accurate and efficient numerical techniques for the solution of the BTE. The discrete ordinates method (DOM) and more recently, the lattice Boltzmann method (LBM) have been used for this purpose. In this work, a comparison between the numerical solution of the phonon BTE by the LBM and DOM is made in order to delineate the strengths and weaknesses of these approaches. Test cases are chosen with Knudsen (Kn) numbers varying between 0.01-100 to cover the full range of diffusive to ballistic phonon transport. The results show that solutions obtained from both methods converge to analytical results for the 1 dimensional phonon transport in a slab. Solutions obtained by two methods converge to analytical solutions of 2 dimensional problems at low Kn. However, solution accuracy is strongly determined by angular resolution for moderate to high Kn. Since the number of propagation directions in LBM are limited, significant errors are engendered in multi-dimensional acoustically-thin problems. DOM also suffers errors at low angular resolutions for high Kn, but yields accurate solutions when sufficient angular resolution is employed.