通过自适应增强RBF-FD在高度非均匀节点分布下的效率
Enhancing RBF-FD Efficiency for Highly Non-Uniform Node Distributions via Adaptivity
AI总结：
提出一种基于局部数据密度的自适应RBF-FD方法，通过多项式细化和自适应模板大小提升微分矩阵稀疏性，在非均匀节点分布下兼顾精度与计算效率。
摘要：
径向基函数生成的有限差分（RBF-FD）方法因其对不规则节点分布的灵活性而近来广受欢迎。然而，文献中的收敛理论在应用于非均匀节点分布时，要求缩小填充距离，且未利用高数据密度区域。使用相同模板大小和附加多项式次数的非自适应方法在高密度区域会有更高的局部精度，但对整体收敛阶没有影响，且可能浪费计算能力。本研究提出一种自适应RBF-FD方法，利用局部数据密度达到理想的阶精度。通过基于数据密度进行多项式细化和使用自适应模板大小，该自适应RBF-FD方法生成的微分矩阵具有更高稀疏性，同时对非均匀点分布实现相同的用户指定收敛阶。这使得该方法能更好地利用高节点密度区域，与标准非自适应RBF-FD方法相比，兼顾了精度与效率。
英文摘要：
Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBF-FD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
