Asymptotically Compatible Fourier Spectral Approximations of Nonlocal Allen--Cahn Equations

Du, Qiang; Yang, Jiang

doi:10.1137/15m1039857

Cited by 70 publications

(59 citation statements)

References 42 publications

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“…Stability tests. For the case ρ δ (|s|) ∈ L 1 (R 2 ), i.e., α ∈ [0, 2), it has been proved in[15] that the steady state solution u * to the NAC equation (1.1) is continuous if ε 2 C δ ≥ 1, where…”

mentioning

confidence: 99%

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

Xiao

et al. 2019

SIAM J. Numer. Anal.

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219

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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.

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confidence: 99%

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

Xiao

et al. 2019

SIAM J. Numer. Anal.

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219

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“…Such models have attracted the attention of community in recent years [8,239,327,405,408]. Algorithmic development and numerical analysis concerning these nonlocal models (including the fractional ones in either space or time or both ) can be found in [6,49,159,150,238,260,394,445]. Related algorithmic studies with respect to different applications were presented in [17].…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

“…For spatially nonlocal phase field models, due to the spread of nonlocal interactions, the phase field variables may no longer be as smooth as their local counterpart. They may also lead to narrower interfacial region and permit singularities across the interface or at the defects [159,239,394]. 7.5.2.…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

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

Feng

2020

Handbook of Numerical Analysis

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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“…In the special case of periodic domains, the multipliers are in fact the eigenvalues of L δ,β (see Section 4.1). Eigenvalues of nonlocal Laplace operators have been recently studied in [12,13]. The work [12] focused on nonlocal eigenvalues and spectral approximations for a 1-D nonlocal Allen-Cahn (NAC) equation with periodic boundary conditions, and showed that these numerical methods are asymptotically compatible in the sense that they provide convergent approximations to both nonlocal and local NAC models.…”

Section: Introductionmentioning

confidence: 99%

“…The work [12] focused on nonlocal eigenvalues and spectral approximations for a 1-D nonlocal Allen-Cahn (NAC) equation with periodic boundary conditions, and showed that these numerical methods are asymptotically compatible in the sense that they provide convergent approximations to both nonlocal and local NAC models. The work [12] provides numerical examples corresponding to specific integral kernels for which the eigenvalues are computed explicitly. A more general method is provided in [13] for the accurate computations of the eigenvalues in 1, 2, and 3 dimensions for a class of radially symmetric kernels.…”

Section: Introductionmentioning

confidence: 99%

Fourier multipliers for nonlocal Laplace operators

Alali

Albin

2019

Applicable Analysis

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Fourier multiplier analysis is developed for peridynamic Laplace operators, which are defined for scalar fields in R n . The focus is on operators L δ,β with compactly supported integral kernels of the form χ B δ (0) (x) 1x β , which are commonly used in peridynamic models [6]. The Fourier multipliers m(ν) of L δ,β are given through an integral representation, which is shown to be well-defined for β < n + 2. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function 2 F 3 in any spatial dimension n and for any β < n + 2. The new general formula for the multipliers is well-defined for any β ∈ R \ {n + 4, n + 6, n + 8, . . .} and is utilized to extend the definition of the peridynamic Laplacians to the case when β ≥ n + 2 (with β = n + 4, n + 6, n + 8, . . .). Some special cases are presented. We show that the Fourier multipliers of L δ,β converge to the Fourier multipliers of ∆ in the limit as β → n + 2. In addition, we show that in the limit as β → n + 2 − , L δ,β u converges to ∆u for sufficiently regular u. Moreover, we identify the limit of the Fourier multipliers of L δ,β as β → −∞ and, furthermore, recognize the limit operator L δ,−∞ as an integral operator with kernel supported in the (n − 1)-sphere.Asymptotic analysis of 2 F 3 is utilized to identify the asymptotic behavior of the Fourier multipliers as ν → ∞. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel (i.e., when β < n), diverge to −∞ at a rate proportional to − log( ν ) when β = n, and diverge to −∞ at a rate proportional to − ν β−n when β > n. The bounds and decay rates are presented explicitly in terms of n, β, and the nonlocality δ.In periodic domains, the Fourier multipliers are the eigenvalues of L δ,β and therefore the presented results provide characterization formulas and asymptotic behavior for the eigenvalues. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation.

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Asymptotically Compatible Fourier Spectral Approximations of Nonlocal Allen--Cahn Equations

Cited by 70 publications

References 42 publications

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

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

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

Fourier multipliers for nonlocal Laplace operators

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