Unconditionally energy stable second-order numerical scheme for the Allen–Cahn equation with a high-order polynomial free energy

In this article, we consider a temporally second-order unconditionally energy stable computational method for the Allen–Cahn (AC) equation with a high-order polynomial free energy potential. By modifying the nonlinear parts in the governing equation, we have a linear convex splitting scheme of the energy for the high-order AC equation. In addition, by combining the linear convex splitting with a strong-stability-preserving implicit–explicit Runge–Kutta (RK) method, the proposed method is linear, temporally second-order accurate, and unconditionally energy stable. Computational tests are performed to demonstrate that the proposed method is accurate, efficient, and energy stable.

A convergent convex splitting scheme for a nonlocal Cahn--Hilliard--Oono type equation with a transport term

We devise a first-order in time convex splitting scheme for a nonlocal Cahn--Hilliard--Oono type equation with a transport term and subject to homogeneous Neumann boundary conditions. However, we prove the stability of our scheme when the time step is sufficiently small, according to the velocity field and the interaction kernel. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which confirm our theoretical results and demonstrate the performance of our scheme not only for phase separation, but also for crystal nucleation, for several choices of the interaction kernel.

Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation

Strong-stability-preserving (SSP) implicit–explicit (IMEX) Runge–Kutta (RK) methods for the Cahn–Hilliard (CH) equation with a polynomial double-well free energy density were presented in a previous work, specifically H. Song’s “Energy SSP-IMEX Runge–Kutta Methods for the Cahn–Hilliard Equation” (2016). A linear convex splitting of the energy for the CH equation with an extra stabilizing term was used and the IMEX technique was combined with the SSP methods. And unconditional strong energy stability was proved only for the first-order methods. Here, we use a nonlinear convex splitting of the energy to remove the condition for the convexity of split energies and give a stability condition for the coefficients of the second-order method to preserve the discrete energy dissipation law. Along with a rigorous proof, numerical experiments are presented to demonstrate the accuracy and unconditional strong energy stability of the second-order method.

A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

An energy-stable pseudospectral scheme for Swift-Hohenberg equation with its Lyapunov functional

We analyze a first order in time Fourier pseudospectral scheme for Swift-Hohenberg equation. One major challenge for the higher order diffusion non-linear systems is how to ensure the unconditional energy stability and we propose an efficient scheme for the equation based on the convex splitting of the energy. The?oretically, the energy stability of the scheme is proved. Moreover, following the derived aliasing error estimate, the convergence analysis in the discrete l2-norm for the proposed scheme is given.