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

TL;DR

This paper introduces a semi-implicit Lax-Wendroff kinetic scheme for simulating hydrodynamic phonon transport in solids, effectively capturing multi-scale thermal conduction phenomena with larger cell sizes and time steps.

Contribution

It develops a novel semi-implicit scheme that couples phonon migration and scattering, improving computational efficiency and accuracy in modeling phonon transport.

Findings

Accurately captures multi-scale thermal conduction phenomena.

Allows larger cell sizes and time steps than phonon mean free path.

Demonstrates effectiveness across different scattering rates.

Abstract

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.