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

TL;DR

This work presents a semi-implicit Lax-Wendroff kinetic scheme for solving the phonon Boltzmann transport equation under the Callaway double-relaxation-time model, unifying ballistic, hydrodynamic, and diffusive heat transport. By using a finite-volume framework with second-order interfacial reconstructions and a split temporal integration (forward Euler for migration and backward Euler for scattering), the method couples migration with both normal and resistive scattering within a single time step, allowing time steps and cell sizes to exceed the phonon mean free path in appropriate regimes. Numerical tests across quasi-1D, transient thermal grating, and quasi-2D in-plane geometries demonstrate accurate replication of analytical solutions and literature data across a wide range of Knudsen numbers, including the hydrodynamic regime and second-sound-like behavior. The approach offers an efficient, robust tool for multi-scale phonon transport modeling with potential for large-scale, multi-physics simulations.

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.

Figures (4)

• Figure 1: Steady spatial distributions of the temperature, where ${T^*} = \left( T-T_L\right)/(T_h-T_L)$, $x^*=x/L$. The solid diamonds present the data obtained from previous references MajumdarA93FilmNanalyticalLIU2022111436. The solid circles and lines present the simulated values. (a,c) $\Delta x=0.1, \Delta t=0.04$. (b) $\Delta x=0.001, \Delta t=0.0004$.
• Figure 2: Temporal attenuation of the temperature fluctuation amplitude across various Knudsen numbers. $\Delta x$=0.02, $\Delta t$=0.005. 'Present' is the simulated value, where $N_{\theta} \times N_{\varphi} = 48 \times 48$, 'Reference' denotes the data obtained from previous papers collins_non-diffusive_2013heatwaves_2022chuangPhysRevB.104.245424. (a) Only R-scattering. (b) Only N-scattering. (c) Both R-scattering and N-scattering.
• Figure 3: (a) $Kn_R=10^{-3}$, $Kn_N=10^5$, $\Delta x$/$\lambda_R$=12.5, $\Delta t$/$\tau_R$=8. Reference solutions in the diffusive regime are obtained from Ref. collins_non-diffusive_2013 (b) $Kn_R=10^5$, $Kn_N=10^{-3}$, $\Delta x$/$\lambda_N$=10, $\Delta t$/$\tau_N$=5. Reference solutions in the hydrodynamics regime are obtained from Ref. heatwaves_2022chuang.
• Figure 4: Spatial profiles of heat flux in the diffusive regime, where $N_{\theta} \times N_{\varphi} = 24 \times 24$, $\Delta y$=0.01, $\Delta t$=0.004, 'Analytical' is the Fuchs-Sondheimer analytical solutions Fuchs_1938_analyticalSondheimer_1952_analytical. 'Reference' is the data obtained from a previous paper LIU2022111436.