Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle

Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle

Year:    2016

Author:    Tao Tang, Jiang Yang

Journal of Computational Mathematics, Vol. 34 (2016), Iss. 5 : pp. 451–461

Abstract

It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1603-m2014-0017

Journal of Computational Mathematics, Vol. 34 (2016), Iss. 5 : pp. 451–461

Published online:    2016-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    11

Keywords:    Allen-Cahn Equations Implicit-explicit scheme Maximum principle Nonlinear energy stability.

Author Details

Tao Tang

Jiang Yang

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  3. A class of monotone and structure-preserving Du Fort-Frankel schemes for nonlinear Allen-Cahn equation

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  4. Convolution tensor decomposition for efficient high-resolution solutions to the Allen–Cahn equation

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  5. Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

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  6. Up to eighth-order maximum-principle-preserving methods for the Allen–Cahn equation

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  7. Lack of robustness and accuracy of many numerical schemes for phase-field simulations

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    https://doi.org/10.1142/S0218202523500409 [Citations: 2]
  8. A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion

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  9. Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation

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    https://doi.org/10.1155/2023/1869660 [Citations: 0]
  10. THE FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE IN A SPACE-FRACTIONAL ALLEN-CAHN EQUATION

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    https://doi.org/10.11948/20230364 [Citations: 0]
  11. High-order $$L^{2}$$-bound-preserving Fourier pseudo-spectral schemes for the Allen-Cahn equation

    Teng, Xueqing | Zhang, Hong

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  12. Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

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    https://doi.org/10.1002/num.22333 [Citations: 13]
  13. A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation

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    https://doi.org/10.1016/j.apnum.2019.12.018 [Citations: 29]
  14. Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme

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    https://doi.org/10.1016/j.amc.2022.127374 [Citations: 4]
  15. Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation

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    https://doi.org/10.1137/18M118236X [Citations: 163]
  16. Stability analysis for a maximum principle preserving explicit scheme of the Allen–Cahn equation

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    https://doi.org/10.1016/j.matcom.2023.01.016 [Citations: 14]
  17. Stabilized Energy Factorization Approach for Allen–Cahn Equation with Logarithmic Flory–Huggins Potential

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  18. Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes

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  19. Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes

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  20. A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

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  21. Transformed Model Reduction for Partial Differential Equations with Sharp Inner Layers

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  22. Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

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  23. An explicit stable finite difference method for the Allen–Cahn equation

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  24. Maximum bound principle preserving linear schemes for nonlocal Allen–Cahn equation based on the stabilized exponential-SAV approach

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  25. Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models

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  26. Gradient-descent-like scheme for the Allen–Cahn equation

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  27. Analysis and numerical methods for nonlocal‐in‐time Allen‐Cahn equation

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  29. Linear energy stable and maximum principle preserving semi-implicit scheme for Allen–Cahn equation with double well potential

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  30. Fourier Spectral High-Order Time-Stepping Method for Numerical Simulation of the Multi-Dimensional Allen–Cahn Equations

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  31. Discrete Maximum Principle and Energy Stability Analysis of Du Fort-Frankel Scheme for 1D Allen-Cahn Equation

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  33. Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation

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  34. Arbitrarily High-Order Maximum Bound Preserving Schemes with Cut-off Postprocessing for Allen–Cahn Equations

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  35. A maximum bound principle preserving iteration technique for a class of semilinear parabolic equations

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  36. Positivity-preserving and unconditionally energy stable numerical schemes for MEMS model

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  37. Linearly Implicit Schemes Preserve the Maximum Bound Principle and Energy Dissipation for the Time-fractional Allen–Cahn Equation

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  39. Stabilized Integrating Factor Runge--Kutta Method and Unconditional Preservation of Maximum Bound Principle

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  40. A linear second-order maximum bound principle preserving finite difference scheme for the generalized Allen–Cahn equation

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  41. A second-order Strang splitting scheme with exponential integrating factor for the Allen–Cahn equation with logarithmic Flory–Huggins potential

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  42. A linear second order unconditionally maximum bound principle-preserving scheme for the Allen-Cahn equation with general mobility

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  43. Generalized SAV-Exponential Integrator Schemes for Allen--Cahn Type Gradient Flows

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  44. A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint

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  45. Arbitrarily High-Order Exponential Cut-Off Methods for Preserving Maximum Principle of Parabolic Equations

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  46. Unconditionally maximum principle-preserving linear method for a mass-conserved Allen–Cahn model with local Lagrange multiplier

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    https://doi.org/10.1016/j.cnsns.2024.108327 [Citations: 0]
  47. Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

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  48. A new high-order maximum-principle-preserving explicit Runge–Kutta method for the nonlocal Allen–Cahn equation

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    https://doi.org/10.1016/j.cam.2023.115500 [Citations: 0]
  49. Maximum Principle Preserving Schemes for Binary Systems with Long-Range Interactions

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  50. Unconditional energy stability and maximum principle preserving scheme for the Allen-Cahn equation

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    https://doi.org/10.1007/s11075-024-01880-2 [Citations: 0]
  51. The fractional Allen–Cahn equation with the sextic potential

    Lee, Seunggyu | Lee, Dongsun

    Applied Mathematics and Computation, Vol. 351 (2019), Iss. P.176

    https://doi.org/10.1016/j.amc.2019.01.037 [Citations: 3]
  52. Unconditionally positivity preserving and energy dissipative schemes for Poisson–Nernst–Planck equations

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    https://doi.org/10.1007/s00211-021-01203-w [Citations: 23]
  53. Neural Control of Parametric Solutions for High-Dimensional Evolution PDEs

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  54. Higher-order energy-decreasing exponential time differencing Runge-Kutta methods for gradient flows

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