基于流守恒和傅立叶谱分析的混合型Wigner-Poisson方程组求解

doi:10.19596/j.cnki.1001-246x.8613

计算物理 ›› 2023, Vol. 40 ›› Issue (2): 248-257.DOI: 10.19596/j.cnki.1001-246x.8613

所属专题：贺贤土院士从事科学研究工作60周年暨激光聚变相关研究进展专刊

• 贺贤土院士从事科学研究工作60周年暨激光聚变相关研究进展专刊 • 上一篇下一篇

基于流守恒和傅立叶谱分析的混合型Wigner-Poisson方程组求解

胡天行¹, 盛正卯¹^,*(), 吴栋²^,*()

1. 浙江大学物理学院, 聚变理论与模拟中心, 浙江杭州 310027
2. 上海交通大学物理与天文学院, 激光等离子体研究所, 惯性约束聚变科学与应用协同创新中心, 上海 200240

收稿日期:2022-08-15 出版日期:2023-03-25 发布日期:2023-07-05
通讯作者: 盛正卯, 吴栋
作者简介:
胡天行, 男, 博士研究生, 研究方向为高能量密度物理中的量子效应
基金资助:
国家自然科学基金(12075204); 中国科学院战略性先导科技专项A类(XDA250050500); 上海市科技创新行动计划(22JC1401500)

Solution of Wigner-Poisson System with Combining Flux Balance and Fourier Spectrum Methods

Tianxing HU¹, Zhengmao SHENG¹^,*(), Dong WU²^,*()

1. Institute of Fusion theory and simulation, School of Physics, Zhejiang University, Hangzhou, Zhejiang 301400, China
2. Collaborative Innovation Center of Inertial Fusion Science and Applications, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

Received:2022-08-15 Online:2023-03-25 Published:2023-07-05
Contact: Zhengmao SHENG, Dong WU

摘要/Abstract

摘要：

提出一种基于流守恒和傅立叶分析的混合型离散欧拉网格法来求解Wigner-Poisson方程组。在坐标和速度空间分别采用不同的时间推进算法。与一般的离散欧拉法相比, 可以显著提高非线性模拟结果的可靠性。通过该方法求解一些常见静电动理学不稳定性在量子等离子体中的行为变化; 通过线性的本征解验证代码的可靠性, 然后进行一些非线性现象的模拟, 包含非线性朗道阻尼以及双流不稳定性的非线性饱和等。

关键词: 量子等离子体, 动理学, 欧拉法, 非线性效应

Abstract:

The difference between quantum plasma and classical plasma is mainly reflected in the following two aspects: 1) the statistical equilibrium state of the system changes from the classical Maxwell distribution to the Fermi-Dirac distribution; 2) the single-particle quantum wave effect of electrons cannot be avoided. Corresponding to the Vlasov equation in the classical plasma, the kinetic equation of the quantum plasma is the Wigner equation, but the numerical solution of the Wigner equation is more complicated than the Vlasov equation. In this paper, we propose a new method based on flux balance and Fourier spectrum methods. The hybrid method is used to solve the Wigner-Poisson equations. This method adopts different time advancing algorithms in the coordinate and velocity spaces. Compared with the general discrete Euler method, it can significantly improve the accuracy of nonlinear simulation results. This paper investigates the behavioral changes of some common electrostatic kinetic instabilities in quantum plasmas through this method; verifies the reliability of the code through linear eigensolutions, and then simulates some nonlinear phenomena, including Nonlinear Landau damping and nonlinear saturation for two-stream instability, etc.

Key words: quantum plasmas, kinetic, Euler method, nonlinear effects

胡天行, 盛正卯, 吴栋. 基于流守恒和傅立叶谱分析的混合型Wigner-Poisson方程组求解[J]. 计算物理, 2023, 40(2): 248-257.

Tianxing HU, Zhengmao SHENG, Dong WU. Solution of Wigner-Poisson System with Combining Flux Balance and Fourier Spectrum Methods[J]. Chinese Journal of Computational Physics, 2023, 40(2): 248-257.

图/表 9

图1 纯傅里叶方法和混合方法计算(a)双流不稳定性的(b)能量守恒过程对比

Fig.1 Comparison of pure Fourier method and Hybrid method (a) two-stream instability; (b) energy conservation

图2 纯傅里叶方法和混合方法计算经典BGK平衡的结果对比(a)FSM($t=187 \omega_{\mathrm{p}}^{-1} $); (b)Hybrid($ t=187 \omega_{\mathrm{p}}^{-1}$)

Fig.2 Comparison of pure Fourier method and Hybrid method of a classical BGK equilibrium (a)FSM($ t=187 \omega_{\mathrm{p}}^{-1}$); (b)Hybrid($t=187 \omega_{\mathrm{p}}^{-1} $)

图3 Recurrence现象的演示(a)没有超碰撞算符的朗道阻尼；(b)加入了超碰撞算符

Fig.3 Demonstration of recurrence (a) Landau damping without hyper-collision; (b) without hyper-collision

图4 (a) 朗道阻尼与(b)双流不稳定性本征值解与线性模拟的对比

Fig.4 Comparison of eigen-value solution and linear simulation of (a) Landau damping and (b) two-stream instability

图5 量子朗道阻尼的色散关系随Q的变化

Fig.5 Quantum Landau damping with varying Q

图6 量子朗道阻尼的色散关系随Θ的变化

Fig.6 Quantum Landau damping with varying Θ

图7 双流不稳定的本征值解与线性模拟解的对比

Fig.7 Comparison of the eigen-value solution and the numerical solution of two-stream instability

图8 经典与量子的非线性朗道阻尼模拟结果

Fig.8 Numerical results of nonlinear Landau damping: classical and quantum mechanical.

图9 相空间中经典与量子BGK平衡态演化过程的对比 (a) t=48ωp-1; (b) t=55ωp-1; (c) t=80ωp-1; (d) t=316ωp-1; (e) t=48ωp-1; (f) t=55ωp-1; (g) t=80ωp-1; (h) t=316ωp-1

Fig.9 Comparison of classical and quantum BGK equilibrium (a) t=48ωp-1; (b) t=55ωp-1; (c) t=80ωp-1; (d) t=316ωp-1; (e) t=48ωp-1; (f) t=55ωp-1; (g) t=80ωp-1; (h) t=316ωp-1

参考文献 28

1	贺贤土. 等离子体中集体效应对韧致辐射的影响[J]. 物理学报, 1981, 30 (11): 1415- 1422.
2	贺贤土. 激励Brillouin散射非线性致稳过程的阻尼效应[J]. 物理学报, 1982, 31 (7): 882- 894. DOI
3	贺贤土. 等离子体中大幅波与低频振荡粒子非线性相互作用效应[J]. 物理学报, 1982, 31 (10): 1317- 1336.
4	贺贤土. 高频波激发低频磁场和离子声波的重整化强湍动理论[J]. 物理学报, 1986, 35 (3): 283- 299.
5	贺贤土. 等离子体中粒子与cavitons相互作用[J]. 物理学报, 1987, 36 (2): 199- 207. DOI
6	ZHU S P , HE X T , ZHENG C . Slow-time-scale magnetic fields driven by fast-time-scale waves in an underdense relativistic Vlasov plasma[J]. Physics of Plasmas, 2001, 8 (1): 321- 328. DOI
7	ZHU S P , ZHENG C Y , HE X T . A theoretical model for a spontaneous magnetic field in intense laser plasma interaction[J]. Physics of Plasmas, 2003, 10 (10): 4166- 4168. DOI
8	HE X , LI J , FAN Z , et al. A hybrid-drive nonisobaric-ignition scheme for inertial confinement fusion[J]. Physics of Plasmas, 2016, 23 (8): 082706. DOI
9	ZHANG J , WANG W , YANG X , et al. Double-cone ignition scheme for inertial confinement fusion[J]. Philosophical Transactions of the Royal Society A, 2020, 378 (2184): 20200015. DOI
10	DORNHEIM T , GROTH S , BONITZ M . The uniform electron gas at warm dense matter conditions[J]. Physics Reports, 2018, 744, 1- 86. DOI
11	MANFREDI G , HERVIEUX P A , HURST J . Phase-space modeling of solid-state plasmas[J]. Reviews of Modern Plasma Physics, 2019, 3 (1): 1- 55. DOI
12	CAI H B , YAN X X , YAO P L , et al. Hybrid fluid-particle modeling of shock-driven hydrodynamic instabilities in a plasma[J]. Matter and Radiation at Extremes, 2021, 6 (3): 035901. DOI
13	WU D , YU W , FRITZSCHE S , et al. Particle-in-cell simulation method for macroscopic degenerate plasmas[J]. Physical Review E, 2020, 102 (3): 033312. DOI
14	BONITZ M . Quantum kinetic theory[M]. Berlin: Springer, 2016.
15	LIANG J H , HU T X , WU D , et al. Kinetic study of quantum two-stream instability by Wigner approach[J]. Physical Review E, 2021, 103 (3): 033207. DOI
16	SUH N-D , FEIX M R , BERTRAND P . Numerical simulation of the quantum Liouville-Poisson system[J]. Journal of Computational Physics, 1991, 94 (2): 403- 418. DOI
17	WIGNER E . On the quantum correction for thermodynamic equilibrium[J]. Physical Review, 1932, 40 (5): 749- 759. DOI
18	KADANOFF L , BAYM G . Quantum statistical mechanics[M]. New York: W A Benjamin Inc, 1962.
19	CHENG C Z , KNORR G . The integration of the Vlasov equation in configuration space[J]. Journal of Computational Physics, 1976, 22 (3): 330- 351. DOI
20	STRANG G . On the construction and comparison of difference schemes[J]. SIAM Journal on Numerical Analysis, 1968, 5 (3): 506- 517. DOI
21	FILBET F , SONNENDRÜCKER E , BERTRAND P . Conservative numerical schemes for the Vlasov equation[J]. Journal of Computational Physics, 2001, 172 (1): 166- 187. DOI
22	BERNSTEIN I B , GREENE J M , KRUSKAL M D . Exact nonlinear plasma oscillations[J]. Physical Review, 1957, 108 (3): 546- 550. DOI
23	ZAKHAROV V E . Collapse of Langmuir waves[J]. Sov Phys JETP, 1972, 35 (5): 908- 914.
24	DANNERT T , JENKO F . Vlasov simulation of kinetic shear Alfvén waves[J]. Computer Physics Communications, 2004, 163 (2): 67- 78. DOI
25	MANFREDI G , HAAS F . Self-consistent fluid model for a quantum electron gas[J]. Physical Review B, 2001, 64 (7): 075316. DOI
26	HAAS F , MANFREDI G , FEIX M . Multistream model for quantum plasmas[J]. Physical Review E, 2000, 62 (2): 2763. DOI
27	MANFREDI G . How to model quantum plasmas[J]. Fields Inst Commun, 2005, 46, 263- 287.
28	DALIGAULT J . On the quantum Landau collision operator and electron collisions in dense plasmas[J]. Physics of Plasmas, 2016, 23 (3): 032706.

[1]	周远志, 郑春阳, 刘占军, 曹莉华, 程瑞锦, 贺贤土. 弱磁化等离子体中受激拉曼散射和受激布里渊散射抑制研究[J]. 计算物理, 2023, 40(2): 136-146.
[2]	张文帅, 蔡洪波, 张恩浩, 杜报, 邹士阳, 朱少平. 马赫数对等离子体碰撞冲击波结构的影响———全动理学模拟研究[J]. 计算物理, 2023, 40(2): 199-209.
[3]	邹兴, 陈松泽, 郭照立. 精细平衡气体动理学格式[J]. 计算物理, 2020, 37(6): 653-666.
[4]	肖敏, 徐喜华, 倪国喜. 基于任意拉格朗日—欧拉型移动网格的反应流广义黎曼问题方法[J]. 计算物理, 2020, 37(2): 127-139.
[5]	曾伟, 陈松泽, 郭照立. 气体动理学格式模拟多孔介质流动的可行性[J]. 计算物理, 2019, 36(5): 551-558.
[6]	陈玺君, 郭照立. 表征体元尺度渗流的离散统一动理学格式[J]. 计算物理, 2019, 36(4): 386-394.
[7]	甄亚欣, 倪国喜. 反应流体的移动网格动理学格式[J]. 计算物理, 2015, 32(6): 677-684.
[8]	徐喜华, 倪国喜. 基于WENO重构的高阶移动网格动理学格式[J]. 计算物理, 2013, 30(4): 509-514.
[9]	仇轶, 由长福, 祁海鹰, 徐旭常. 无网格方法中的背景积分方案及单颗粒下降过程的数值模拟[J]. 计算物理, 2006, 23(5): 525-529.
[10]	陆慧林, 赵广播, 刘文铁, 别如山. 低颗粒浓度气固两相均匀剪切流[J]. 计算物理, 1999, 16(4): 436-441.
[11]	段文山, 吕克朴, 赵金保. Toda晶格中特定孤立子的碰撞[J]. 计算物理, 1994, 11(2): 129-133.

基于流守恒和傅立叶谱分析的混合型Wigner-Poisson方程组求解

Solution of Wigner-Poisson System with Combining Flux Balance and Fourier Spectrum Methods

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

图/表 9

参考文献 28

相关文章 11

编辑推荐

Metrics

作者中心

审稿中心

期刊浏览

期刊介绍