A hierarchical method for verification of particle-in-cell/ Monte Carlo collision modelling on plasma discharges
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摘要: 当前，科学计算的验证主要针对基于确定性偏微分方程组的网格离散方法。放电等离子体的粒子云网格PIC方法作为一种粒子-网格耦合的仿真手段，其验证方法具有显著不同的特点：第一，PIC仿真除了在时间和空间上进行离散，还需要对粒子数权重进行离散；第二，离散粒子的相空间分布函数是否适合作为验证研究的观测量；第三，粒子-网格耦合过程中的电场插值和电荷分配会影响PIC仿真的全局收敛精度。另外，当PIC方法与蒙特卡罗（MC）方法耦合时，离散误差和随机误差通常叠加在一起，理查德森外推需要结合系综平均进行。提出了一种分层级验证的方法。首先对单粒子轨道、电磁场求解、二体粒子碰撞进行收敛精度阶测试；然后采用空间电荷限制流、气体的傅里叶流动等具有精确解的经典物理模型分别对集成PIC、MC模块进行离散误差评估；最后采用放电物理过程对程序功能进行基准校验。Abstract: The verification of scientific computing currently places a strong emphasis on grid discretization methods for systems of deterministic partial differential equations. However, verifying particle-in-cell (PIC) simulations, which employ a particle-mesh method to model discharging plasmas, presents distinctive challenges. Firstly, PIC simulations require discretization not only in time and space but also in macro-particle weights. Secondly, challenges arise regarding the utilization of the discretized particle phase space distribution function for verification purposes. Thirdly, the interpolation of electric fields and charge distribution can significantly impact the overall accuracy of PIC convergence.When PIC methods are integrated with Monte Carlo (MC) methods, discretization and stochastic errors often combine, necessitating Richardson extrapolation in conjunction with ensemble averaging.To tackle these challenges, this paper introduces a hierarchical verification approach. It commences with order-of-accuracy tests for individual particle trajectories, electromagnetic field solvers, and binary-particle collisions. Discretization errors for integrated PIC and MC modules are then evaluated using classical physical models that possess exact solutions, such as space-charge-limited current and Fourier flow in gases. Finally, a code-to-code comparison is performed with benchmark examples of simplified discharge simulations.
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Key words:
- PIC method /
- verification techniques /
- order of accuracy test /
- discharge simulation
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图 1 自编程序的粒子推进Boris算法和odeint的计算结果比较
Figure 1. Comparison of calculation results for self-implemented particle push using Boris algorithm and odeint
图 2 BKW气体的速度矩与时间的关系：数据点是使用蒙特卡罗碰撞方法计算的，而曲线是理论值
Figure 2. Velocity moments of BKW gas versus time: data points are calculated with Monte Carlo collision method and line is from theory
图 3 空间和时间步长固定时，热通量随宏粒子权重的变化
Figure 3. Variation of heat flux with macro-particle weight when spatial and temporal steps are fixed
图 4 空间步长和宏粒子权重固定时，热通量、离散误差、统计误差随无量纲时间步长的变化
Figure 4. Variation of heat flux, discretization error, and statistical error with dimensionless time step when spatial step and macro-particle weight are fixed
图 5 离散网格一致细化时热通量随广义网格间距的变化
Figure 5. Variation of heat flux with generalized grid spacing during grid refinement with discrete grid uniformity
图 6 PIC计算的SCLC密度与朱氏模型的比较
Figure 6. Comparison of SCLC density calculated by PIC and Zhu’s model
图 7 简化反应通道模型气体击穿的粒子模拟基准校验：与商业软件对比宏粒子数随时间的变化
Figure 7. Benchmark validation of particle simulation for simplified reaction channel model of gas breakdown: comparison of macro-particle count vs time with commercial software
表 1 粒子模拟验证的第二层级功能验证
Table 1. Second-level functional verification of particle simulation validation
function typical case particle push particle collision field solvers gather and scatter particle boundaries DSMC Fourier heat flow × × O O reflective boundary MCC charged particle transport × × O O O PIC space charge limited current × O × × absorptive boundary 
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表 2 带电粒子在等离子体中的输运
Table 2. Transport of charged particles in a plasma
physical problem typical case collision type transport of electrons in weakly
ionized plasmasMaxwell gas model electron-heavy particle elastic collisions model/hard sphere gas electron-heavy particle elastic + excitation Lucas-Salee model electron-heavy particle elastic + excitation + ionization Ness-Robson model electron-heavy particle elastic + excitation + ionization + attachment ion transport in weakly
ionized plasmaion velocity distribution in an
alternating electric fieldelastic collisions between ions and heavy particles transport in fully ionized plasma Spitzer conductivity Coulomb collisions between electrons and ions
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