用于加速稀疏网格粒子-in-单元方法中粒子-网格耦合的精确层次算法
Exact hierarchical algorithms for accelerating particle--mesh coupling in sparse-grid particle-in-cell methods
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中文总结 AI 辅助
本文提出两种适用于SGCT-PIC和HSG-PIC的精确层次算法，将粒子-网格耦合复杂度降低，二维实验获多倍加速且保持精度。
中文摘要 AI 辅助
本文提出两种适用于稀疏网格组合技术粒子-in-单元（SGCT-PIC）和层次稀疏网格粒子-in-单元（HSG-PIC）方法的电荷沉积与电场插值层次算法。这两种算法受快速多极子方法（FMM）启发，利用与粒子填充框的有向无环图（DAG）相关的粒子簇，减少粒子-网格相互作用的数量。粒子-网格相互作用由分段多项式核控制，因此相关的多极展开是精确的，无需截断或近似，且在近场和远场区域均有效，从而消除了多极到局部转换的需求。电荷沉积和场插值步骤的算术复杂度从 $Ø(p^d n^{d-1}N)$ 降低至 $Ø(p^d(N+M))$，其中 $M=2^{dn}$ 表示全网格节点数，在考虑的区域中通常不大于粒子总数，即 $M\lesssim N$。二维配置下的数值实验表明，对于SGCT-PIC，电荷沉积加速比为8.2倍至66.9倍，对于HSG-PIC为3.1倍至18.8倍；场插值加速比分别为4.1倍至62.6倍和4.2倍至13.7倍，具体取决于每单元粒子数，同时保持精确的粒子-网格相互作用。加速比随每单元粒子数增加而增大，反映出层次算法对粒子数量的依赖性降低，且在大粒子群体中优势愈发显著。
英文摘要
In this paper, we propose two hierarchical algorithms for charge deposition and electric-field interpolation that apply to both the sparse-grid combination technique (SGCT-PIC) and hierarchical sparse-grid (HSG-PIC) particle-in-cell methods. The two algorithms are inspired by the fast multipole method (FMM) and exploit clusters of particles associated with a directed acyclic graph (DAG) of particle-populated boxes to reduce the number of particle--mesh interactions. The particle--mesh interactions are governed by piecewise-polynomial kernels, so that the associated multipole expansions are exact, requiring neither truncation nor approximation, and are valid in both near- and far-field regions, thereby eliminating the need for multipole-to-local translations. The arithmetic complexity of the charge deposition and field interpolation steps is reduced from $Ø(p^d n^{d-1}N)$ to $Ø(p^d(N+M))$, where $M=2^{dn}$ denotes the number of full-grid mesh nodes and is typically no larger than the particle population in the considered regime, $M\lesssim N$. Numerical experiments in two-dimensional configurations demonstrate charge-deposition speedups of $8.2\times$--$66.9\times$ for SGCT-PIC and $3.1\times$--$18.8\times$ for HSG-PIC, and field-interpolation speedups of $4.1\times$--$62.6\times$ and $4.2\times$--$13.7\times$, respectively, depending on the particle-per-cell ratio, while preserving the exact particle--mesh interactions. The speedups increase with the particle-per-cell ratio, reflecting the reduced dependence of the hierarchical algorithms on the number of particles and their increasing advantage for large particle populations.
