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.

翻译：径向基函数生成有限差分（RBF-FD）方法因其在非规则节点分布下的灵活性而近年来备受关注。然而，现有文献中的收敛理论在应用于非均匀节点分布时，要求填充距离持续缩减，且未能充分利用高数据密度区域的优势。采用相同模板尺寸与附属多项式次数的非自适应方法虽能在高密度区域获得较高局部精度，但对整体收敛阶数并无提升，且可能造成计算资源的浪费。本文提出一种自适应RBF-FD方法，通过利用局部数据密度实现所需阶数的精度。该方法基于数据密度进行多项式细化并自适应调整模板尺寸，从而在非均匀点分布下生成具有更高稀疏性的微分矩阵，同时达到用户指定的相同收敛阶数。相较于标准非自适应RBF-FD方法，该自适应方法能更有效地利用高节点密度区域的优势，在保持精度的同时提升计算效率。