High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model

Abstract

AbstractWe propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.