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Second-order diffusion limit for the phonon transport equation-asymptotics and numerics

Anjali Nair, Qin Li, Weiran Sun

TL;DR

This work addresses deriving and validating a second-order diffusion limit for the phonon transport equation in the small-Knudsen-number regime. The authors separate interior and boundary-layer contributions, derive a Robin-type boundary condition for the macroscopic diffusion equation, and introduce a spectral half-space solver to compute the needed coefficients. Key contributions include a second-order asymptotic expansion, a frequency-averaged treatment of Knudsen numbers, and a practical numerical pipeline that achieves quadratic accuracy in Kn while handling both incoming and reflective boundaries. The results demonstrate improved accuracy over leading-order diffusion models and provide a scalable approach for simulating boundary-influenced phonon-mediated heat transfer.

Abstract

We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate.

Second-order diffusion limit for the phonon transport equation-asymptotics and numerics

TL;DR

This work addresses deriving and validating a second-order diffusion limit for the phonon transport equation in the small-Knudsen-number regime. The authors separate interior and boundary-layer contributions, derive a Robin-type boundary condition for the macroscopic diffusion equation, and introduce a spectral half-space solver to compute the needed coefficients. Key contributions include a second-order asymptotic expansion, a frequency-averaged treatment of Knudsen numbers, and a practical numerical pipeline that achieves quadratic accuracy in Kn while handling both incoming and reflective boundaries. The results demonstrate improved accuracy over leading-order diffusion models and provide a scalable approach for simulating boundary-influenced phonon-mediated heat transfer.

Abstract

We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate.
Paper Structure (7 sections, 2 theorems, 52 equations, 8 figures, 1 algorithm)

This paper contains 7 sections, 2 theorems, 52 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Equation and its asymptotic limit
  3. Phonon transport equation
  4. Classical asymptotic derivation
  5. Boundary layer correction
  6. Computation of the diffusion limit
  7. Numerical Results

Key Result

Theorem 1

Let $f$ satisfy the Cauchy problem with the equation eqn:f, and assume $\mathsf{Kn}\to 0$ (or $L\to\infty$), then in this limiting regime, $f(x,v,\omega)\to\rho(x)$ that satisfies: Moreover, one has $f(x,v,\omega)=\rho(x)-v\mathsf{Kn}\partial_x\rho +O(\mathsf{Kn}^2)$.

Figures (8)

  • Figure 1: Example I. Reference solution for $\mathsf{Kn}=0.0625$.
  • Figure 2: Example I. The panel on the left shows the density $\rho$ over the whole domain. The panel in the middle shows the layer behavior close to $x=0$ computed using different $\mathsf{Kn}$ and the limiting $\rho$. The panel on the right shows the convergence rate on the log-log scale. It suggests the asymptotic convergence is $\mathsf{Kn}^2$. The incoming data is $\phi = v$.
  • Figure 3: Example I. The panel on the left shows the density $\rho$ over the whole domain. The panel in the middle shows the layer behavior close to $x=0$ computed using different $\mathsf{Kn}$ and the limiting $\rho$. The panel on the right shows the convergence rate on the log-log scale. It suggests the asymptotic convergence is $\mathsf{Kn}^2$. The incoming data is $\phi = v^2$.
  • Figure 4: Example I. If Dirichlet boundary condition is used for the limiting equation, first order convergence is obtained.
  • Figure 5: Reference solution for $\mathsf{Kn}=0.0625$ in the multi-frequency case when $\phi=v$.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Remark 2
  • Theorem 3