Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations
Numerical Analysis
2026-05-05 v3 Numerical Analysis
Abstract
This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including a quasi-Fermat theorem and a quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its second-order consistency and convergence. By modifying this scheme, we also obtain a numerical scheme with zero local truncation error for ODEs whose solution trajectories are ellipses.
Cite
@article{arxiv.2601.09900,
title = {Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations},
author = {Kiyuob Jung},
journal= {arXiv preprint arXiv:2601.09900},
year = {2026}
}