A third-order fully decoupled relaxed E-SAV scheme for the hydrodynamically-coupled binary phase-field crystal model

Yue Shen; Wenjie Wang; Xiaoyue Xie; Yue Shen; Wenjie Wang; Xiaoyue Xie

doi:10.3934/math.2026954

AIMS Mathematics

2026, Volume 11, Issue 8: 23687-23717. doi: 10.3934/math.2026954

Previous Article Next Article

Research article

A third-order fully decoupled relaxed E-SAV scheme for the hydrodynamically-coupled binary phase-field crystal model

1.
School of Science, Xi'an University of Architecture and Technology, Xi'an 710055, China
2.
Equipment Management and Unmanned Aerial Vehicle Engineering School, Air Force Engineering University, Xi'an 710051, China

Received: 09 May 2026 Revised: 23 July 2026 Accepted: 24 July 2026 Published: 03 August 2026
MSC : 35Q30, 65M06, 65M12, 76D05

This paper develops a third-order fully decoupled numerical scheme for the hydrodynamically-coupled binary phase-field crystal (BPFC) model based on the relaxed exponential scalar auxiliary variable (E-SAV) approach. The governing equations are derived from the $ L^2 $-gradient flow and coupled with the incompressible Navier-Stokes equations. To preserve total mass conservation, two nonlocal Lagrange multipliers are introduced into the chemical potentials. For temporal discretization, a third-order backward differentiation formula (BDF3) combined with explicit extrapolation techniques is adopted, leading to a linear and fully decoupled time-marching scheme. In addition, a relaxation strategy is incorporated into the E-SAV framework to improve the consistency between the modified energy and the original energy functional. Compared with existing second-order relaxed E-SAV schemes, the proposed method achieves higher temporal accuracy while retaining the advantages of linearity, decoupling, and energy stability. The proposed scheme only requires solving several linear elliptic equations with constant coefficients at each time step, making the algorithm computationally efficient and easy to implement. The unique solvability and discrete energy dissipation property of the numerical scheme are theoretically established. Furthermore, a consistency analysis is provided to show that the relaxation step preserves the third-order temporal accuracy of the primary variables and does not affect the overall convergence order of the scheme. Numerical experiments verify the expected third-order convergence rate, demonstrate the unconditional energy stability of the proposed scheme, and illustrate its effectiveness for long-time simulations of hydrodynamically-coupled crystal growth problems.
- hydrodynamically-coupled BPFC model,
- relaxed E-SAV method,
- BDF3 scheme,
- fully decoupled algorithm,
- energy stability
Citation: Yue Shen, Wenjie Wang, Xiaoyue Xie. A third-order fully decoupled relaxed E-SAV scheme for the hydrodynamically-coupled binary phase-field crystal model[J]. AIMS Mathematics, 2026, 11(8): 23687-23717. doi: 10.3934/math.2026954

Related Papers:

Abstract

This paper develops a third-order fully decoupled numerical scheme for the hydrodynamically-coupled binary phase-field crystal (BPFC) model based on the relaxed exponential scalar auxiliary variable (E-SAV) approach. The governing equations are derived from the $ L^2 $-gradient flow and coupled with the incompressible Navier-Stokes equations. To preserve total mass conservation, two nonlocal Lagrange multipliers are introduced into the chemical potentials. For temporal discretization, a third-order backward differentiation formula (BDF3) combined with explicit extrapolation techniques is adopted, leading to a linear and fully decoupled time-marching scheme. In addition, a relaxation strategy is incorporated into the E-SAV framework to improve the consistency between the modified energy and the original energy functional. Compared with existing second-order relaxed E-SAV schemes, the proposed method achieves higher temporal accuracy while retaining the advantages of linearity, decoupling, and energy stability. The proposed scheme only requires solving several linear elliptic equations with constant coefficients at each time step, making the algorithm computationally efficient and easy to implement. The unique solvability and discrete energy dissipation property of the numerical scheme are theoretically established. Furthermore, a consistency analysis is provided to show that the relaxation step preserves the third-order temporal accuracy of the primary variables and does not affect the overall convergence order of the scheme. Numerical experiments verify the expected third-order convergence rate, demonstrate the unconditional energy stability of the proposed scheme, and illustrate its effectiveness for long-time simulations of hydrodynamically-coupled crystal growth problems.

References

[1]	K. R. Elder, M. Katakowski, M. Haataja, M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701. https://doi.org/10.1103/PhysRevLett.88.245701 doi: 10.1103/PhysRevLett.88.245701
[2]	K. R. Elder, M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004), 051605. https://doi.org/10.1103/PhysRevE.70.051605 doi: 10.1103/PhysRevE.70.051605
[3]	S. M. Wise, C. Wang, J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269–2288. https://doi.org/10.1137/080738143 doi: 10.1137/080738143
[4]	A. Baskaran, Z. Z. Hu, J. S. Lowengrub, C. Wang, S. M. Wise, P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, J. Comput. Phys., 250 (2013), 270–292. https://doi.org/10.1016/j.jcp.2013.04.024 doi: 10.1016/j.jcp.2013.04.024
[5]	J. Shen, J. Xu, J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407–416. https://doi.org/10.1016/j.jcp.2017.10.021 doi: 10.1016/j.jcp.2017.10.021
[6]	J. Shen, J. Xu, Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows, SIAM J. Numer. Anal., 56 (2018), 2895–2912. https://doi.org/10.1137/17m1159968 doi: 10.1137/17m1159968
[7]	X. F. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294–316. https://doi.org/10.1016/j.jcp.2016.09.029 doi: 10.1016/j.jcp.2016.09.029
[8]	Z. G. Liu, X. L. Li, Efficient modified techniques of invariant energy quadratization approach for gradient flows, Appl. Math. Lett., 98 (2019), 206–214. https://doi.org/10.1016/j.aml.2019.06.006 doi: 10.1016/j.aml.2019.06.006
[9]	J. S. Kou, S. Y. Sun, X. H. Wang, A novel energy factorization approach for the diffuse-interface model with Peng-Robinson equation of state, SIAM J. Sci. Comput., 42 (2020), B30–B56. https://doi.org/10.1137/19m1251230 doi: 10.1137/19m1251230
[10]	X. H. Wang, J. S. Kou, J. C. Cai, Stabilized energy factorization approach for Allen-Cahn equation with logarithmic Flory-Huggins potential, J. Sci. Comput., 82 (2020), 25. https://doi.org/10.1007/s10915-020-01127-x doi: 10.1007/s10915-020-01127-x
[11]	X. H. Wang, J. S. Kou, H. C. Gao, Linear energy stable and maximum principle preserving semi-implicit scheme for Allen-Cahn equation with double well potential, Commun. Nonlinear Sci. Numer. Simul., 98 (2021), 105766. https://doi.org/10.1016/j.cnsns.2021.105766 doi: 10.1016/j.cnsns.2021.105766
[12]	B. H. Xia, C. L. Mei, Q. Yu, Y. B. Li, A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect, Comput. Methods Appl. Mech. Eng., 363 (2020), 112795. https://doi.org/10.1016/j.cma.2019.112795 doi: 10.1016/j.cma.2019.112795
[13]	H. G. Lee, J. Shin, J. Y. Lee, First and second order operator splitting methods for the phase field crystal equation, J. Comput. Phys., 299 (2015), 82–91. https://doi.org/10.1016/j.jcp.2015.06.038 doi: 10.1016/j.jcp.2015.06.038
[14]	Y. B. Li, J. Kim, An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation, Comput. Methods Appl. Mech. Eng., 319 (2017), 194–216. https://doi.org/10.1016/j.cma.2017.02.022 doi: 10.1016/j.cma.2017.02.022
[15]	C. Wang, S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945–969. https://doi.org/10.1137/090752675 doi: 10.1137/090752675
[16]	H. G. Lee, J. Shin, J. Y. Lee, First- and second-order energy stable methods for the modified phase field crystal equation, Comput. Methods Appl. Mech. Eng., 321 (2017), 1–17. https://doi.org/10.1016/j.cma.2017.03.033 doi: 10.1016/j.cma.2017.03.033
[17]	Z. G. Liu, X. L. Li, Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation, Numer. Algorithms, 85 (2020), 107–132. https://doi.org/10.1007/s11075-019-00804-9 doi: 10.1007/s11075-019-00804-9
[18]	M. Wang, Q. M. Huang, C. Wang, A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation, J. Sci. Comput., 88 (2021), 33. https://doi.org/10.1007/s10915-021-01487-y doi: 10.1007/s10915-021-01487-y
[19]	X. L. Li, J. Shen, Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation, Sci. China Math., 65 (2022), 2201–2218. https://doi.org/10.1007/s11425-020-1867-8 doi: 10.1007/s11425-020-1867-8
[20]	J. W. Wu, J. X. Yang, Z. J. Tan, Highly efficient and fully decoupled BDF time-marching schemes with unconditional energy stabilities for the binary phase-field crystal models, Eng. Comput., 39 (2023), 3157–3181. https://doi.org/10.1007/s00366-022-01727-1 doi: 10.1007/s00366-022-01727-1
[21]	X. Zhang, J. W. Wu, Z. J. Tan, Highly efficient, decoupled and unconditionally stable numerical schemes for a modified phase-field crystal model with a strong nonlinear vacancy potential, Comput. Math. Appl., 132 (2023), 119–134. https://doi.org/10.1016/j.camwa.2022.12.011 doi: 10.1016/j.camwa.2022.12.011
[22]	H. G. Lee, A new conservative Swift-Hohenberg equation and its mass conservative method, J. Comput. Appl. Math., 375 (2020), 112815. https://doi.org/10.1016/j.cam.2020.112815 doi: 10.1016/j.cam.2020.112815
[23]	Z. J. Tan, L. Chen, J. X. Yang, Generalized Allen-Cahn-type phase-field crystal model with FCC ordering structure and its conservative high-order accurate algorithm, Comput. Phys. Commun., 286 (2023), 108656. https://doi.org/10.1016/j.cpc.2023.108656 doi: 10.1016/j.cpc.2023.108656
[24]	Q. Li, N. Cui, S. P. Zheng, L. Q. Mei, A new Allen-Cahn type two-model phase-field crystal model for FCC ordering and its numerical approximation, Appl. Math. Lett., 132 (2022), 108211. https://doi.org/10.1016/j.aml.2022.108211 doi: 10.1016/j.aml.2022.108211
[25]	J. An, J. Zhang, X. F. Yang, A novel second-order time accurate fully discrete finite element scheme with decoupling structure for the hydrodynamically-coupled phase field crystal model, Comput. Math. Appl., 113 (2022), 70–85. https://doi.org/10.1016/j.camwa.2022.01.029 doi: 10.1016/j.camwa.2022.01.029
[26]	X. F. Yang, X. M. He, Numerical approximations of flow coupled binary phase field crystal system: fully discrete finite element scheme with second-order temporal accuracy and decoupling structure, J. Comput. Phys., 467 (2022), 111448. https://doi.org/10.1016/j.jcp.2022.111448 doi: 10.1016/j.jcp.2022.111448
[27]	J. X. Yang, J. Kim, Consistent energy-stable method for the hydrodynamics coupled PFC model, Int. J. Mech. Sci., 241 (2023), 107952. https://doi.org/10.1016/j.ijmecsci.2022.107952 doi: 10.1016/j.ijmecsci.2022.107952
[28]	J. W. Wu, X. Zhang, Y. Y. Wu, Z. J. Tan, Consistency-enhanced E-SAV BDF2 time-marching method with relaxation for the hydrodynamically-coupled binary phase-field crystal model, Commun. Nonlinear Sci. Numer. Simul., 145 (2025), 108730. https://doi.org/10.1016/j.cnsns.2025.108730 doi: 10.1016/j.cnsns.2025.108730
[29]	H. L. Liao, Y. Y. Kang, $ L^2 $ norm error estimates of BDF methods up to fifth-order for the phase field crystal model, IMA J. Numer. Anal., 44 (2024), 2138–2164. https://doi.org/10.1093/imanum/drad047 doi: 10.1093/imanum/drad047

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)