Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations

TL;DR

This paper introduces specular differentiation and develops numerical schemes for first-order ODEs, demonstrating their effectiveness through analysis and simulations, including a scheme with zero local truncation error for elliptical solutions.

Contribution

It proposes specular differentiation and constructs novel numerical schemes with proven convergence and a zero local truncation error for specific ODE solutions.

Findings

Developed a second-order convergent numerical scheme for first-order ODEs.

Proved a zero local truncation error scheme for elliptical solution trajectories.

Validated schemes through numerical simulations.

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.