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We explore a class of splitting schemes employing implicit-explicit (IMEX) time-stepping to achieve accurate and energy-stable solutions for thin-film equations and Cahn-Hilliard models with variable mobility. This splitting method incorporates a linear, constant coefficient implicit step, facilitating efficient computational implementation. We investigate the influence of stabi- lizing splitting parameters on the numerical solution computationally, considering various initial conditions. Furthermore, we generate energy-stability plots for the proposed methods, examin- ing different choices of splitting parameter values and timestep sizes. These methods enhance the accuracy of the original bi-harmonic-modified (BHM) approach, while preserving its energy- decreasing property and achieving second-order accuracy. We present numerical experiments to illustrate the performance of the proposed methods.
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Efficient Variable Time-stepping Adaptive DLN Algorithms for the Allen-Cahn Equation
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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- Award ID(s):
- 2309590
- PAR ID:
- 10633215
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Journal of Scientific Computing
- Volume:
- 104
- Issue:
- 2
- ISSN:
- 0885-7474
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s10915-025-02980-4
