AI 中文总结
该研究针对分层稀疏网格粒子-in-单元方法提出带能量逼近空间与分层余量策略的局部自适应方案，可在减少网格节点和粒子数量的同时提升局部解逼近精度，为复杂结构等离子体高效模拟提供了可行思路。
AI 中文摘要
本文针对分层稀疏网格粒子-in-单元（HSG-PIC）方法引入了新的逼近空间和局部自适应加密策略，以在保留稀疏网格方法降噪特性的同时改善偏差。我们首先提出一种基于能量的逼近空间，该空间优化了$\text{H}^1$-范数误差与自由度数量之间的关系，同时提出一类广义稀疏网格空间，可连续衔接经典稀疏网格与全网格逼近。随后，我们开发了一种基于分层余量的局部自适应逼近策略，并结合高效的增量加密算法，避免在完整广义空间上求解伽辽金问题。数值实验表明，所提出的自适应HSG-PIC方法大幅提升了对具有局部结构的解的逼近精度，同时保留了稀疏网格离散化的统计优势。与标准全网格PIC方法相比，该自适应方法在显著减少网格节点和粒子数量的情况下，达到了相当或更高的精度。这些结果证明了自适应稀疏网格PIC方法在高效模拟具有复杂解结构的动力学等离子体方面的潜力。
英文摘要
In this paper, we introduce new approximation spaces and a locally adaptive refinement strategy for the hierarchical sparse-grid PIC (HSG-PIC) method to improve the bias while preserving the noise-reduction properties of sparse-grid methods. We first propose an energy-based approximation space, which optimizes the relation between the $\mathrm{H}^1$-norm error and the number of degrees of freedom, together with a family of generalized sparse-grid spaces that continuously connects classical sparse-grid and full-grid approximations. We then develop a locally adaptive approximation strategy based on hierarchical surpluses, combined with an efficient incremental refinement algorithm that avoids solving the Galerkin problem on the complete generalized space. Numerical experiments demonstrate that the proposed adaptive HSG-PIC method substantially improves the approximation of solutions with localized structures while maintaining the statistical advantages of sparse-grid discretizations. Compared with a standard full-grid PIC method, the adaptive approach achieves comparable or higher accuracy with a significantly reduced number of mesh nodes and particles. These results demonstrate the potential of adaptive sparse-grid PIC methods for efficient simulations of kinetic plasmas with complex solution structures.
