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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.
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This content will become publicly available on January 1, 2026
Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow
Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard (FCH) energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the “morphological complexity” observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The re- sult is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split di- rectly into networks of interfaces. We evaluate a number of schemes that use second or- der backward differentiation formula (BDF2) type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a preconditioned steepest descent (PSD) nonlin- ear solver or upon linearly implicit time discretization based on the standard implicit- explicit (IMEX) and the scalar auxiliary variable (SAV) approaches. We add an ex- ponential time differencing (ETD) scheme for comparison purposes.
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- PAR ID:
- 10618074
- Publisher / Repository:
- Global Science
- Date Published:
- Journal Name:
- Communications in Computational Physics
- Volume:
- 37
- Issue:
- 4
- ISSN:
- 1815-2406
- Page Range / eLocation ID:
- 877 to 920
- Subject(s) / Keyword(s):
- Phase field model benchmark computations adaptive time stepping functionalized Cahn-Hilliard
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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