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

TL;DR

This paper derives a second-order accurate limiting equation for phonon transport in the small Knudsen number regime and provides a spectral method-based numerical approach to compute the associated Robin coefficients, validated through numerical experiments.

Contribution

It introduces a second-order asymptotic limit for the phonon transport equation and a spectral method for computing Robin coefficients via an auxiliary half-space problem.

Findings

Second-order convergence of the limiting equation demonstrated.

Spectral method effectively computes Robin coefficients.

Numerical results confirm the theoretical asymptotic rate.

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.