An unconditionally stable scheme for the Allen–Cahn equation with high-order polynomial free energy

2021

Adv Differ Equ

Self Cite

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.

Section: Effect Of P On the Interfacial Dynamics In 3dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kim

2021

Adv Differ Equ

Self Cite

“…The CS scheme is handed down to implement the hAC equation in an arbitrary order; however, studying the effective time step on the slow dynamics that occur as the order increases is necessary. 19 In fact, the CS scheme for the AC equation ( 4) can be interpreted as a semi-implicit scheme with different time step scaling as follows:…”

Section: Introductionmentioning

confidence: 99%

Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

Yoon

Numerical Meth Engineering

et al. 2022

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An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high‐order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi‐implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum‐principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high‐order potential is alleviated by the time step modification through the results of motion by mean curvature.

A linear convex splitting scheme for the Cahn–Hilliard equation with a high‐order polynomial free energy

Numerical Meth Engineering

Yoon

Kim

2023

Self Cite

In this article, we present an unconditionally energy stable linear scheme for the Cahn–Hilliard equation with a high‐order polynomial free energy. The classical Cahn–Hilliard equation does not satisfy the maximum principle; hence the order parameter can be shifted out of the minimum values of the double‐well potential. We adopt a high‐order polynomial potential to diminish this effect and employ the efficient linear convex splitting scheme. Since the stabilizing factor gradually increases as the degree of potential becomes greater, we modify a non‐physical part of potential as a fourth‐order polynomial to reduce the stabilizing factor. Numerical results as well as theoretical results demonstrate the accuracy and energy stability of our method. Furthermore, we verify that some limitations arising from applications of the classical Cahn–Hilliard model can be resolved by adopting a high‐order free energy.