Non-stiff methods for Airy flow and the modified Korteweg-de Vries equation

TL;DR

This paper develops and analyzes non-stiff numerical methods for simulating the evolution of 2-D curves under Airy flow, where curvature follows the modified Korteweg-de Vries equation, demonstrating their accuracy and stability.

Contribution

It introduces fully discrete space-time methods for Airy flow with rigorous convergence analysis and numerical validation, advancing computational techniques for dispersive geometric evolutions.

Findings

Methods are accurate and stable for simulating Airy flow.

Convergence of the numerical schemes is rigorously proven.

Numerical experiments confirm efficiency and reliability.

Abstract

In this paper, we implement non-stiff interface tracking methods for the evolution of 2-D curves that follow Airy flow, a curvature-dependent dispersive geometric evolution law. The curvature of the curve satisfies the modified Korteweg-de Vries equation, a dispersive non-linear soliton equation. We present a fully discrete space-time analysis of the equations (proof of convergence) and numerical evidence that confirms the accuracy, convergence, efficiency, and stability of the methods.