Efficient energy-stable second-order predictor-corrector SAV scheme for the Cahn-Hilliard equation: algorithm, analysis, and computation

Shimin Lin; Chi Fui William Ni; Jun Zhang; Pengtao Yue; Shimin Lin; Chi Fui William Ni; Jun Zhang; Pengtao Yue

doi:10.3934/era.2025278

Electronic Research Archive

2025, Volume 33, Issue 10: 6298-6321. doi: 10.3934/era.2025278

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Efficient energy-stable second-order predictor-corrector SAV scheme for the Cahn-Hilliard equation: algorithm, analysis, and computation

1.
School of Science, Jimei University, Xiamen 361021, China
2.
The Pennington School, Pennington, NJ 08534, USA
3.
Computational Mathematics Research Center, Guizhou University of Finance and Economics, Guiyang 550025, China
4.
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Received: 30 August 2025 Revised: 22 September 2025 Accepted: 09 October 2025 Published: 24 October 2025

The Cahn-Hilliard equation plays a central role in modeling phase separation processes in complex systems, including alloys, polymers, and biological materials. Numerical schemes for this equation must balance efficiency, stability, and accuracy in order to capture the rich dynamics of interfacial evolution. In this work, we developed a new second-order predictor–corrector scheme within the scalar auxiliary variable (SAV) framework, combined with Crank–Nicolson (CN) time discretization. The proposed method is linear, uniquely solvable, and unconditionally energy stable, while also providing rigorous error estimates. Computational experiments demonstrated that the new scheme not only maintains second-order temporal accuracy for relatively large time steps, but also yields smaller numerical errors compared to standard SAV-CN methods. These results highlight both the theoretical advantages and practical potential of the predictor–corrector SAV approach for advancing accurate and efficient simulations of phase-field models in science and engineering.
- SAV,
- IEQ,
- predictor-corrector,
- time-discrete,
- second-order,
- error estimates
Citation: Shimin Lin, Chi Fui William Ni, Jun Zhang, Pengtao Yue. Efficient energy-stable second-order predictor-corrector SAV scheme for the Cahn-Hilliard equation: algorithm, analysis, and computation[J]. Electronic Research Archive, 2025, 33(10): 6298-6321. doi: 10.3934/era.2025278

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Abstract

The Cahn-Hilliard equation plays a central role in modeling phase separation processes in complex systems, including alloys, polymers, and biological materials. Numerical schemes for this equation must balance efficiency, stability, and accuracy in order to capture the rich dynamics of interfacial evolution. In this work, we developed a new second-order predictor–corrector scheme within the scalar auxiliary variable (SAV) framework, combined with Crank–Nicolson (CN) time discretization. The proposed method is linear, uniquely solvable, and unconditionally energy stable, while also providing rigorous error estimates. Computational experiments demonstrated that the new scheme not only maintains second-order temporal accuracy for relatively large time steps, but also yields smaller numerical errors compared to standard SAV-CN methods. These results highlight both the theoretical advantages and practical potential of the predictor–corrector SAV approach for advancing accurate and efficient simulations of phase-field models in science and engineering.

References

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