A differentiation formula, with application to the two-dimensional Schr\"odinger equation

TL;DR

This paper introduces a new differentiation formula based on generalized divided differences, improving the numerical solution of the 2D Schrödinger equation by achieving faster convergence and higher accuracy.

Contribution

The paper presents a novel differentiation method using generalized divided differences and demonstrates its effectiveness in solving the 2D Schrödinger eigenvalue problem.

Findings

Enhanced accuracy in eigenvalue computations

Faster convergence compared to standard methods

Effective application to 2D Schrödinger equation

Abstract

A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent variable. This method is applied to the numerical solution of the eigenvalue problem for the two-dimensional Schr\"odinger equation, where standard methods converge very slowly while the approach proposed here gives accurate results.