This work introduces an approach to variable-step Finite Difference Method (FDM) where non-uniform meshes are generated via a weight function, which establishes a diffeomorphism between uniformly spaced computational coordinates and variably spaced physical coordinates. We then derive finite difference approximations for derivatives on variable meshes in both one-dimensional and multi-dimensional cases, and discuss constraints on the weight function. To demonstrate efficacy, we apply the method to the two-dimensional time-independent Schr\"odinger equation for a harmonic oscillator, achieving improved eigenfunction resolution without increased computational cost.

翻译：本文提出了一种变步长有限差分法（FDM）的实现途径，其中非均匀网格通过权重函数生成，该函数在均匀分布的计算坐标与变间距的物理坐标之间建立了微分同胚映射。我们随后推导了一维及多维情形下变网格上导数的有限差分近似格式，并讨论了权重函数的约束条件。为验证方法的有效性，我们将该方法应用于二维谐振子的定态薛定谔方程，在不增加计算成本的前提下实现了本征函数分辨率的提升。