[Paper Review] A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional Allen-Cahn equations

Dongdong He, Kejia Pan+1 more|arXiv (Cornell University)|Apr 19, 2018

Solidification and crystal growth phenomena1 references4 citations

TL;DR

This paper proposes a fourth-order maximum principle preserving operator splitting scheme for the three-dimensional space fractional Allen-Cahn equation. By combining Strang splitting, Crank-Nicolson ADI for the fractional diffusion step, fourth-order finite differences, and Richardson extrapolation, the method achieves unconditional stability and preserves the discrete maximum principle under a time step constraint, with numerical validation of fourth-order accuracy and maximum principle satisfaction in both 2D and 3D tests.

ABSTRACT

In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically verified.

Motivation & Objective

• To develop a high-order numerical scheme for the three-dimensional space fractional Allen-Cahn equation that preserves the discrete maximum principle.
• To achieve fourth-order temporal and spatial accuracy while maintaining unconditional stability.
• To ensure the numerical solution remains bounded within [−1, 1] under a reasonable time step constraint.
• To provide a linear, efficient, and unconditionally stable method that avoids nonlinear solvers or energy-stability trade-offs.
• To numerically verify the preservation of the discrete maximum principle and the scheme's high-order convergence.

Proposed method

• Strang’s second-order operator splitting is applied to decompose the fractional Allen-Cahn equation into three sub-steps: two involving analytical ODEs and one involving a 3D linear fractional diffusion equation.
• The 3D fractional diffusion step is solved using the Crank-Nicolson alternating direction implicit (ADI) method, which reduces the multidimensional problem to a sequence of one-dimensional solves.
• Fourth-order finite difference schemes are used to discretize the space-fractional derivatives, ensuring high-order spatial accuracy.
• Richardson extrapolation is applied to the time integration to elevate the temporal accuracy to fourth order.
• Fourier analysis is used to prove unconditional stability of the overall scheme for smooth solutions.
• A time step constraint is derived to ensure the discrete maximum principle is preserved numerically.

Experimental results

Research questions

• RQ1 Can a high-order (fourth-order) numerical scheme be constructed for the 3D space fractional Allen-Cahn equation that preserves the discrete maximum principle?
• RQ2 Is it possible to achieve both high-order temporal and spatial accuracy while maintaining unconditional stability in the presence of fractional derivatives?
• RQ3 Does the proposed linear operator splitting scheme preserve the solution bounds [−1, 1] under a reasonable time step constraint?
• RQ4 How does the fractional order α affect the interface thickness and coarsening dynamics in phase separation simulations?
• RQ5 Can Richardson extrapolation effectively enhance the temporal accuracy of a splitting scheme without compromising stability or maximum principle preservation?

Key findings

• The proposed scheme is unconditionally stable, as proven by Fourier analysis, for smooth solutions of the 3D fractional Allen-Cahn equation.
• The method achieves fourth-order accuracy in both time and space variables, as confirmed by convergence tests with fabricated smooth solutions.
• The discrete maximum principle is numerically verified: the solution remains bounded within [−1, 1] when the time step satisfies the derived constraint (e.g., Δt ≤ 0.4945 for α=1.7).
• When the time step exceeds the stability-bound constraint (e.g., Δt = 2), the maximum value of the solution exceeds 1, violating the discrete maximum principle.
• Reducing the fractional order α leads to thinner interfaces and slower phase coarsening, with α = 1.2 producing the most heterogeneous and refined phase structures.
• The scheme successfully captures complex 3D phase separation dynamics, including interface evolution and coarsening, with high fidelity and bounded solutions.