Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation

Abstract

The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadraturebased finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

The phase field method for geometric moving interfaces and their numerical approximations

“…Discussion on nonlocal interactions in the form of integral operators may be tracked back to the work of Van der Waals [409], see discussions made in [348], The usual differential equation form of the local phase field energy can be derived from the nonlocal version via the so-called Landau expansion [296], assuming a smooth and spatially slowly varying field. A number of studies on nonlocal Allen-Cahn and nonlocal Cahn-Hilliard can be found in [51,48,55,109,108,148,214,275]. More studies on nonlocal modeling, analysis and computation can be found in [146].…”

Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

“…This is an important and well-known property of the standard CS scheme of the classical Allen-Cahn equation. More investigations on maximum principle preserving schemes for integer order Allen-Cahn equations can be found in [16,17,44,47]. Theorem 3.1 The CS scheme (3.2) satisfies the discrete maximum principle unconditionally, i.e.,…”

“…Classical phase-field models are diffuse interface models that have found numerous applications in diverse research areas, e.g., hydrodynamics [4,33,40], material sciences [2,9], biology [18,42,51] and image processing [38,52], to name just a few. Recently, there are also many studies on nonlocal phase field models involving spatially nonlocal interactions [3,[5][6][7]15,16,22,23,45,49], see [13,14] for more extensive reviews of the literature. Historically, nonlocal interactions in phase field models expressed mathematically in terms of integral operators have been noted in the work of van der Waals [50], see discussions made in [39].…”

Efficient numerical methods for phase-field equations

“…高阶 ETD 格式与相应的数值结果参见文献 [45]. [42,44,45] 中的讨论. 而 对于一般的区域和矩阵 L, 计算 ϕ-函数与向量乘积的常用算法有 Laplace 逆变换法、Taylor 级数法和 Krylov 子空间法等, 更多的算法以及对各个算法计算效果的评述可参见文献 [47,48]…”

“…在非局部 Cahn-Hilliard 方程的数值计算中, 非线性凸分裂格式 [96,97] 、SSI 格式 [14] 和 IEQ 格式 [98] 都给出很好 的计算结果. 非局部 Allen-Cahn 方程满足类似于经典 Allen-Cahn 方程的最大模上界原理, 文献 [44] 给出了保持最大模上界原理的一阶和二阶 ETD 格式, 并证明了能量稳定性和收敛性. 为了减少相场模型大时间数值计算中的时间开销, 通常可以采用自适应方法来加速计算过程.…”