An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm
An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm
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DOI：
10.1016/j.jcp.2011.05.031
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发表时间：
2011-08-01
影响因子：
4.1
通讯作者：
Barnes, D. C.
中科院分区：
文献类型：
--
作者：
Chen, G.;Chacon, L.;Barnes, D. C.
本文讨论了一种新的全隐式的一维静电粒子等离子体模拟方法。与早期的隐式静电PIC方法（基于线性化的Vlasov-Poisson公式）不同，我们的方法基于非线性收敛的Vlasov-Ampere（VA）模型。通过迭代粒子和场到严格的非线性收敛容限，该方法具有上级稳定性和精度特性，避免了早期隐式PIC实现中的大多数精度陷阱。特别是，制定是稳定的时间（Courant-Friedrichs-Lewy）和空间（混叠）的不稳定性。对于任意隐式时间步长，它是电荷和能量守恒的数值舍入（不像早期的“能量守恒”显式PIC公式，它只在任意小的时间步长的限制下保存能量）。虽然动量不完全守恒，但自适应粒子子步进轨道积分器使误差保持较小，这有助于防止粒子隧穿（对长期精度的有害影响）。VA模型是沿沿着粒子轨道的轨道平均，以执行具有粒子子步进的能量守恒定理。因此，非常大的时间步长，仅受感兴趣的动态时间尺度的约束，是可能的，而不损失精度。在数学上，该方法具有Jacobian自由牛顿-克雷洛夫求解器。在这项研究中的一个主要发展是新的时间粒子变量（位置和速度）的非线性消除。这种非线性消除，我们称之为粒子奴役，结果在非线性配方与内存的要求相比，流体计算，并为我们提供了大量的自由度方面的粒子轨道积分。数值例子证明了该计划的广告属性。特别是，长时间的离子声波模拟表明，数值精度不会降低，即使非常大的隐式时间步长，显着的CPU增益是可能的。(C)2011 Elsevier Inc. All rights reserved.
This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov-Poisson formulation), ours is based on a nonlinearly converged Vlasov-Ampere (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant-Friedrichs-Lewy) and spatial (aliasing) instabilities. It is charge- and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier "energy-conserving" explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton-Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible. (C) 2011 Elsevier Inc. All rights reserved.