skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Asymptotic Behaviour of Time Stepping Methods for Phase Field Models
Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ σ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ ϵ → 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity.  more » « less
Award ID(s):
1813203
PAR ID:
10282683
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Journal of Scientific Computing
Volume:
86
Issue:
3
ISSN:
0885-7474
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, Journal of Scientific Computing)
    Abstract We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditionally non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time-adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time-adaptive DLN methods. 
    more » « less
  2. ; (, Discrete & Continuous Dynamical Systems - B)
    In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns. 
    more » « less
  3. ; (, Annals of mathematics)
    The Allen–Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang–Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show that for generic metrics on a 3-manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity one-conjecture and the index lower bound conjecture of Marques–Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen–Cahn min-max constructions were recently carried out by Guaraco and Gaspar–Guaraco. Our resolution of the multiplicity-one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie–Marques–Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every p = 1, 2, 3,…, a two-sided embedded minimal surface with Morse index p and area ~ p13, as conjectured by Marques-Neves. 
    more » « less
  4. ; ; (, Computers & Mathematics with Applications)
    In this paper, we propose a linear second-order numerical method for solving the Allen-Cahn equation with general mobility. The fully-discrete scheme is achieved by using the Crank-Nicolson formula for temporal integration and the central difference method for spatial approximation, together with two additional stabilization terms. Under mild constraints on the two stabilizing parameters, the proposed numerical scheme is shown to unconditionally preserve the discrete maximum bound principle and the discrete original energy dissipation law. Error estimate in the 𝐿∞ norm is successfully derived for the proposed scheme. Finally, some numerical experiments are conducted to verify the theoretical results and demonstrate the performance of the proposed scheme in combination with an adaptive time-stepping strategy. 
    more » « less
  5. ; ; (, Journal of Scientific Computing)
    In contrast to the classical Allen-Cahn equation, the conservative Allen-Cahn equation with a nonlocal Lagrange multiplier not only satisfies the maximum bound principle (MBP) and energy dissipation law but also ensures mass conservation. Many existing schemes often fail to preserve all these properties at the discrete level or require high regularity in time on the exact solution for convergence analysis. In this paper, we construct a new class of low regularity integrators (LRIs) for time discretization of the conservative Allen-Cahn equation by repeatedly using Duhamel's formula. The proposed first- and second-order LRI schemes are shown to conserve mass unconditionally and satisfy the MBP under some time step size constraints. Temporal error estimates for these schemes are derived under a low regularity assumption that the exact solution is only Lipschitz continuous in time, followed by a rigorous proof for energy stability of the corresponding time-discrete solutions. Various numerical experiments and comparisons in two and three dimensions are presented to verify the theoretical results and illustrate the performance of the LRI schemes, especially when the interfacial parameter approaches zero. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email