A Second-Order, Weakly Energy-Stable Pseudo-spectral Scheme for the Cahn–Hilliard Equation and Its Solution by the Homogeneous Linear Iteration Method

Abstract

We present a second order energy stable numerical scheme for the two and three dimensional Cahn---Hilliard equation, with Fourier pseudo-spectral approximation in space. A convex splitting treatment assures the unique solvability and unconditional energy stability of the scheme. Meanwhile, the implicit treatment of the nonlinear term makes a direct nonlinear solver impractical, due to the global nature of the pseudo-spectral spatial discretization. We propose a homogeneous linear iteration algorithm to overcome this difficulty, in which an $$O(s^2)$$O(s2) (where s the time step size) artificial diffusion term, a Douglas---Dupont-type regularization, is introduced. As a consequence, the numerical efficiency can be greatly improved, since the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be implemented with the help of the FFT in a pseudo-spectral setting. Moreover, a careful nonlinear analysis shows a contraction mapping property of this linear iteration, in the discrete $$\ell ^4$$l4 norm, with discrete Sobolev inequalities applied. Moreover, a bound of numerical solution in $$\ell ^\infty $$lź norm is also provided at a theoretical level. The efficiency of the linear iteration solver is demonstrated in our numerical experiments. Some numerical simulation results are presented, showing the energy decay rate for the Cahn---Hilliard flow with different values of $$\varepsilon $$ź.