A Diffeomorphic Variable-Step Finite Difference Method

Abstract: 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.