A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times

Liu，Jia1; Zhang，Chuang1; Yuan，Haizhuan2; Su，Wei3; Wu，Lei1

2022-10-15

10.1016/j.jcp.2022.111436

JOURNAL OF COMPUTATIONAL PHYSICS   Impact Factor & Quartile

0021-9991

1090-2716

The phonon Boltzmann transport equation with dual relaxation times is often used to describe the heat conduction in semiconductor materials when the classical Fourier's law is no longer valid. For practical engineering designs, accurate and efficient numerical methods are highly demanded to solve the equation. At a large Knudsen number (i.e., the ratio of the phonon mean free path to a characteristic system length), steady-state solutions can be obtained via the conventional iterative scheme (CIS) within a few iterations. However, when the Knudsen number becomes small, i.e., when the phonon transport occurs in the diffusive or hydrodynamic regime, thousands of iterations are required to obtain converged results. In this work, a general synthetic iterative scheme (GSIS) is proposed to tackle the inefficiency of CIS. The key ingredient of the GSIS is that a set of macroscopic synthetic equations, which is exactly derived from the Boltzmann transport equation, is simultaneously solved with the kinetic equation to obtain the temperature and heat flux. During the iteration, the macroscopic quantities are used to evaluate the equilibria in the scattering terms of the kinetic equation, thus guiding the evolution of the phonon distribution function, while the distribution function, in turn, provides closures to the synthetic equations. The Fourier stability analysis is conducted to reveal the superiority of the GSIS over the CIS in terms of fast convergence in periodic systems. It is shown that the convergence rate of the GSIS can always be maintained under 0.2 so that only two iterations are required to reduce the iterative error by one order of magnitude. Numerical results in wall-bounded systems are presented to demonstrate further the efficiency of GSIS, where the CPU time is reduced by up to three orders of magnitude, especially in both the diffusive and hydrodynamic regimes where the Knudsen number is small.

Fast convergence Fourier stability analysis General synthetic iterative scheme Phonon Boltzmann equation

EI ; SCI

English

First

National Natural Science Foundation of China[11871414];National Natural Science Foundation of China[12147122];China Postdoctoral Science Foundation[2021M701565];

Computer Science ; Physics

Computer Science, Interdisciplinary Applications ; Physics, Mathematical

WOS:000917225500010

ACADEMIC PRESS INC ELSEVIER SCIENCE

20222912385452

Boltzmann equation ; Distribution functions ; Fourier transforms ; Heat conduction ; Heat flux ; Hydrodynamics ; Integral equations ; Iterative methods ; Kinetic energy ; Kinetic theory ; Kinetics ; Numerical methods ; Phonons ; Semiconductor materials

Fluid Flow, General:631.1 ; Heat Transfer:641.2 ; Semiconducting Materials:712.1 ; Calculus:921.2 ; Mathematical Transformations:921.3 ; Numerical Methods:921.6 ; Statistical Methods:922 ; Probability Theory:922.1 ; Classical Physics; Quantum Theory; Relativity:931

PHYSICS

2-s2.0-85134326965

Scopus

1.Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology,Shenzhen,518055,China
2.School of Mathematics and Computational Science,Xiangtan University,Xiangtan,411105,China
3.School of Engineering,The University of Edinburgh,Edinburgh,EH9 3FD,United Kingdom

Liu，Jia,Zhang，Chuang,Yuan，Haizhuan,et al. A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2022,467.

Liu，Jia,Zhang，Chuang,Yuan，Haizhuan,Su，Wei,&Wu，Lei.(2022).A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times.JOURNAL OF COMPUTATIONAL PHYSICS,467.

Liu，Jia,et al."A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times".JOURNAL OF COMPUTATIONAL PHYSICS 467(2022).