Unconditional energy-stable method for the Swift–Hohenberg equation over arbitrarily curved surfaces with second-order accuracy

Abstract

This paper presents an unconditional energy-stable method for the Swift-Hohenberg equation over arbitrarily curved surfaces. We directly define the Laplace–Beltrami operator on the triangular mesh and its dual mesh, which are the discretizations of regular surface. The direct discretization method has several advantages including intrinsic geometry, convergence property, and mass conservation. A second-order temporal accuracy method has been proposed with the Crank-Nicolson type formulation. We have applied the stabilized splitting method to solve the governing system. Our proposed scheme is easy to implement since the two elliptic equations are both linear with low computational burden. The proposed scheme is provable to be Unconditional energy-stable. Several computational tests are conducted to demonstrate that the numerical method is efficient.