标题： 通过等分布进行变量步长有限差分的实用方法 显示英文标题

标题： A practical recipe for variable-step finite differences via equidistribution

摘要： 我们描述了一个简短且可重复的工作流程，用于在由正权函数g确定的非均匀网格上应用有限差分。该网格通过等分布获得，通过累积积分$S(x)=\int_a^x\!1/g(s)\,ds$及其逆变换将统一的计算坐标$\xi\in[0,1]$映射到物理空间，在多维情况下则通过相应的变扩散（调和）映射及张量$P=(1/g)I$进行映射。然后，我们在不均匀间距上使用标准的三点中心模板计算一阶和二阶导数。我们收集了公式，说明了对g的温和约束（正性、有界性、可积性），并提供了一个小型参考实现。最后，我们在均匀网格和变网格上求解二维定态薛定谔方程，展示了解决局部本征函数的预期改进，而无需增加矩阵大小。我们希望这篇笔记作为一种操作指南，而不是一种新方法，将广泛使用的想法整合成一个单一的、即用型配方，不声称具有新颖性。薛定谔方程，实现了在不增加计算成本的情况下改进本征函数分辨率。 显示英文摘要

摘要： We describe a short, reproducible workflow for applying finite differences on nonuniform grids determined by a positive weight function g. The grid is obtained by equidistribution, mapping uniform computational coordinates $\xi\in[0,1]$ to physical space by the cumulative integral $S(x)=\int_a^x\!1/g(s)\,ds$ and its inverse, and in multiple dimensions by the corresponding variable-diffusion (harmonic) mapping with tensor $P=(1/g)I$. We then use the standard three-point central stencils on uneven spacing for first and second derivatives. We collect the formulas, state the mild constraints on g (positivity, boundedness, integrability), and provide a small reference implementation. Finally, we solve the 2D time-independent Schr\"odinger equation for a harmonic oscillator on uniform vs. variable meshes, showing the expected improvement in resolving localized eigenfunctions without increasing matrix size. We intend this note as a how-to reference rather than a new method, consolidating widely used ideas into a single, ready-to-use recipe, claiming no novelty.ent Schr\"odinger equation for a harmonic oscillator, achieving improved eigenfunction resolution without increased computational cost.