  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.
