A practical recipe for variable-step finite differences via equidistribution

Abstract

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.