Convergence of a finite difference scheme for the Kuramoto--Sivashinsky equation defined on an expanding circle

TL;DR

This paper develops and analyzes a finite difference scheme with Crank--Nicolson for the Kuramoto--Sivashinsky equation on an expanding circle, providing theoretical guarantees and error estimates to aid understanding of solution behaviors.

Contribution

It introduces a new numerical scheme for the Kuramoto--Sivashinsky equation on an expanding circle, with proven convergence, uniqueness, and second-order error estimates.

Findings

The scheme is proven to converge and produce unique solutions.

Second-order accuracy of the numerical method is established.

The approach offers insights into wavenumber selection in related interfacial equations.

Abstract

This paper presents a finite difference method combined with the Crank--Nicolson scheme of the Kuramoto--Sivashinsky equation defined on an expanding circle (\cite{KUY}), and the existence, uniqueness, and second-order error estimate of the scheme. The equation can be obtained as a perturbation equation from the circle solution to an interfacial equation and can provide guidelines for understanding the wavenumber selection of solutions to the interfacial equation. Our proposed numerical scheme can help with such a mathematical analysis.