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

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.