Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations

Ju, Lili; Zhang, Jian; Du, Qiang

doi:10.1016/j.commatsci.2015.04.046

Cited by 75 publications

(63 citation statements)

References 30 publications

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“…Such an advantage leads to some successful applications of ETD schemes on phase field models which usually yield highly stiff ODE systems after suitable spatial discretizations. Ju et al developed stable and compact ETD schemes and their fast implementations for Allen-Cahn [29,49], Cahn-Hilliard [28], and elastic bending energy models [44] by utilizing suitable linear splitting techniques. All the proposed ETD schemes are explicit and thus highly efficient for practical implementations.…”

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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205

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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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205

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“…We then design, implement, and test accurate and efficient numerical methods for solving the gradient-flow equation. Our methods couple a linear splitting scheme [12,24,39,41,42], spectral discretization schemes, and exponential time differencing Runge-Kutta approximations [9,23,26,39]. We finally apply our model and numerical methods to some charged molecules, such a single ion and a two-plate system, demonstrating that our proposed new model performs numerically better than the pervious ones by achieving the force localization near the solute-solvent interface and maintaining more robustly the desirable hyperbolic tangent profile for even larger interfacial width.…”

Section: Introductionmentioning

confidence: 97%

A new phase-field approach to variational implicit solvation of charged molecules with the Coulomb-field approximation

Zhao

Sun

et al. 2018

Communications in Mathematical Sciences

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We construct a new phase-field model for the solvation of charged molecules with a variational implicit solvent. Our phase-field free-energy functional includes the surface energy, solute-solvent van der Waals dispersion energy, and electrostatic interaction energy that is described by the Coulomb-field approximation, all coupled together self-consistently through a phase field. By introducing a new phase-field term in the description of the solute-solvent van der Waals and electrostatic interactions, we can keep the phase-field values closer to those describing the solute and solvent regions, respectively, making it more accurate in the free-energy estimate. We first prove that our phase-field functionals Γ-converge to the corresponding sharp-interface limit. We then develop and implement an efficient and stable numerical method to solve the resulting gradient-flow equation to obtain equilibrium conformations and their associated free energies of the underlying charged molecular system. Our numerical method combines a linear splitting scheme, spectral discretization, and exponential time differencing Runge-Kutta approximations. Applications to the solvation of single ions and a two-plate system demonstrate that our new phase-field implementation improves the previous ones by achieving the localization of the system forces near the solute-solvent interface and maintaining more robustly the desirable hyperbolic tangent profile for even larger interfacial width. This work provides a scheme to resolve the possible unphysical feature of negative values in the phase-field function found in the previous phase-field modeling (cf. H. Sun, et al. J. Chem. Phys., 2015) of charged molecules with the Poisson-Boltzmann equation for the electrostatic interaction. *

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“…Recent years, the fast compact exponential time difference method [9,10,13,5] have compiled much numerical techniques such as matrix diagonal decomposition, linear operator splitting, Fast Fourier Fransformation and so on. At the time of maintaining the high precision and stability, it effectively reduces the computational complexity of the algorithm and reduces the computational cost that the complicated parabolic equations with stiffness can be solved effciently.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, since the first GPU (Graphics Processing Unit) based on CUDA (Computed Unified Device Architecture) was released by NVIDIA in 2006, GPU acceleration technology has been rapidly developed and widely used [9,10,13,5] in general computing. At the moment, the computing performance of high-end GPU has been reached to a trillion times per second.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

GPU-accelerated Exponential Time Difference Method for Parabolic Equations with Stiffness

Xie¹,

Zhu²

2017

Proceedings of the 2017 2nd International Conference on Machinery, Electronics and Control Simulation (MECS 2017)

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Abstract:In this work, we present a Graphics Processing Unit (GPU) accelerated Runge-Kutta exponential time difference (RKETD) method for parabolic equations with stiffness based on CUDA (Computed Unified Device Architecture). In the proposed method, the cuFFT library developed by NVIDIA is employed to compute discrete Fast Fourier Transforms (FFTs) which is the key part in the RKETD method on GPU. Several CUDA code optimization skills are used to improve the speedup performance. The comparison of the numerical results of CUDA-implemented RKETD method on GPU and original RKETD method on CPU demonstrates that the former can obtain good speedup performance. Numerical experiments demonstrate effectiveness and accuracy of the GPU acceleration for typical nonlinear parabolic problem with stiffness.

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Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations

Cited by 75 publications

References 30 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

A new phase-field approach to variational implicit solvation of charged molecules with the Coulomb-field approximation

GPU-accelerated Exponential Time Difference Method for Parabolic Equations with Stiffness

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