Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

Guan, Zhen; Lowengrub, John; Wang, Cheng; Wise, Steven M.

doi:10.1016/j.jcp.2014.08.001

Cited by 154 publications

(94 citation statements)

References 65 publications

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“…⇤ which is obtained by choosing q = 0 andq = sin(⇡x) in (5), for which the estimate is again of the same form as in (47). The results presented in Figure 8 and Table 7 are obtained for the moving interface test case of 1D Cahn-Hilliard equation for 100 time steps with t = 0.00001 under uniform refinement, and confirm the expected consistency of the estimator.…”

Section: Remark 42supporting

confidence: 62%

“…For time discretization, we use the backward Euler method for the heat equation and a first-order semi-implicit splitting scheme from [18] for Cahn-Hilliard equation. Recent second-order time schemes for Cahn-Hilliard models can be found in [46,47]. In the numerical experiments, the time step is chosen su ciently small for both of the equations, in order to avoid time errors due to time discretization.…”

Section: ⌅ 4 Numericsmentioning

confidence: 99%

“…The discrete estimate (47) can be computed at each time step and summed up for the final time, T . For the simulation of the Cahn-Hilliard equation, we choose q 1 = " 2 e u , q 2 = e µ and q = e(T ) as in (39) and we take " = 0.0625 and P e = 1.…”

Section: The Convective Cahn-hilliard Equationmentioning

confidence: 99%

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Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

Şimşek

Zee

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator.

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Section: Remark 42supporting

confidence: 62%

Section: ⌅ 4 Numericsmentioning

confidence: 99%

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Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

Şimşek

Zee

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php error of the spectral approximation of nonlocal models), then high regularity assumptions on u δ are often required. The paper [24] focuses on the energy stability and convergence analysis for the NAC and Cahn-Hilliard equations with a fixed integrable kernel function for fully discrete schemes. Second, even if one can prove the convergence of numerical approximations of nonlocal models with respect to a fixed horizon parameter δ, the convergence behavior may be dependent on δ.…”

Section: Introductionmentioning

confidence: 99%

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

Du¹,

Yang²

2016

SIAM J. Numer. Anal.

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We study Fourier spectral approximations of a nonlocal Allen-Cahn (NAC) equation that reduces to the conventional Allen-Cahn equation in the local limit. We show that the Fourier spectral methods are asymptotically compatible in the sense that they provide convergent approximations to both nonlocal and local models. Furthermore, we provide various error estimates. In particular, it is shown that the numerical solutions of nonlocal models converge to those of the corresponding local models uniformly at a rate of O(δ 2 ). This is achieved by first establishing a similar result for linear nonlocal diffusion equations. A careful investigation, both analytically and computationally, is made of the steady state solutions of NAC equations, demonstrating how discontinuities may appear in solutions and how they are related to model parameters.

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“…In more detail, a convex splitting numerical scheme, which treats the terms of the variational derivative implicitly or explicitly according to whether the terms corresponding to the convex or concave parts of the energy, was formulated in [19], with a mixed finite element approximation in space. Such a numerical approach assures two mathematical properties: unique solvability and unconditional energy stability; also see the related works for various PDE systems, including the phase field crystal (PFC) equation [4,5,27,34,35,39], epitaxial thin film growth model [8,10,31,33], and others [21,22]. Moreover, for a gradient system coupled with fluid motion, the idea of convex splitting can still be applied and these distinguished mathematical properties are retained, as given by a few recent works [9,12,13,19,38].…”

Section: Definition 11 Definementioning

confidence: 99%

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

et al. 2016

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We present and analyze a mixed finite element numerical scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320-1343, 2012), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard ∞ (0, T ; L 2 ) ∩ 2 (0, T ; H 2 ) error estimate, we perform a discrete ∞ (0, T ; H 1 ) ∩ 2 (0, T ; H 3 ) error estimate for the phase variable, through an L 2 inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step τ in terms of the spatial resolution h) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo-Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian h of the numerical solution, such that h φ ∈ S h , for every φ ∈ S h , where S h is the finite element space. B Steven M. Wise

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Second order convex splitting schemes for periodic nonlocal Cahn–Hilliard and Allen–Cahn equations

Cited by 154 publications

References 65 publications

Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

Duality-based two-level error estimation for time-dependent PDEs: Application to linear and nonlinear parabolic equations

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

Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

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