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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.
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A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation
We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy is minimized under a volume constraint and a spanning condition on the level sets of the densities. We discuss two singular limits of these Allen--Cahn Plateau problems: when , we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume ; and when , and , we prove convergence to the classical Plateau problem (in the homotopic spanning formulation of Harrison and Pugh). As a corollary of our analysis we resolve the incompatibility between Plateau's laws and the Allen--Cahn equation implied by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show that Plateau-type singularities can be approximated by energy minimizing solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a transmission condition on a spanning free boundary.
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- PAR ID:
- 10598349
- Publisher / Repository:
- arxiv
- Date Published:
- Edition / Version:
- 2312.11139
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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