A Novel Lagrange Multiplier Approach with Relaxation for Gradient Flows

Abstract

In this paper, we propose a novel Lagrange multiplier approach, named zero-factor (ZF) approach to solve a series of gradient flow problems. The numerical schemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove that the ZF schemes with specific zero factors lead to the popular SAV-type method. To reduce the computation cost and improve the accuracy and consistency, we propose a zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF) method, to design unconditional energy stable schemes for gradient flows. The RZF schemes can be proved to be unconditionally energy stable with respect to a modified energy that is closer to the original energy, and provide a very simple calculation process. The variation of the introduced zero factor is highly consistent with the nonlinear free energy which implies that the introduced ZF method is a very efficient way to capture the sharp dissipation of nonlinear free energy. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

Keywords

Lagrange multiplier approach / zero-factor approach / gradient flows / relaxation / energy stable / numerical examples

References

[1] M. Ashour, N. Valizadeh, and T. Rabczuk, Isogeometric analysis for a phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields, Comput. Methods Appl. Mech. Engrg., 377:113669, 2021.

[2] Q. Cheng, C. Liu, and J. Shen, A new Lagrange multiplier approach for gradient flows, Comput. Methods Appl. Mech. Engrg., 367:113070, 2020.

[3] Q. Cheng and J. Shen, Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model, SIAM J. Sci. Comput., 40:A3982-A4006, 2018.

[4] Q. Cheng and J. Shen, Global constraints preserving scalar auxiliary variable schemes for gradient flows, SIAM J. Sci. Comput., 42:A2489-A2513, 2020.

[5] Q. Du, L. Ju, X. Li, and Z. Qiao, Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation, SIAM J. Numer. Anal., 57(2):875-898, 2019.

[6] Q. Du, L. Ju, X. Li, and Z. Qiao, Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes, SIAM Rev., 63:317-359, 2021.

[7] D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, MRS Online Proceedings Library, 529:39-46, 1998.

[8] Y. Gong, J. Zhao, and Q. Wang, Arbitrarily high-order unconditionally energy stable schemes for thermodynamically consistent gradient flow models, SIAM J. Sci. Comput., 42(1):B135-B156, 2020.

[9] F. Huang, J. Shen, and Z. Yang, A highly efficient and accurate new scalar auxiliary variable approach for gradient flows, SIAM J. Sci. Comput., 42(4):A2514-A2536, 2020.

[10] M. Jiang, Z. Zhang, and J. Zhao, Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation, J. Comput. Phys., 456:110954, 2022.

[11] X. Li, J. Shen, and H. Rui, Energy stability and convergence of SAV block-centered finite difference method for gradient flows, Math. Comp., 88(319):2047-2068, 2019.

[12] L. Lin, Z. Yang, and S. Dong, Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable, J. Comput. Phys., 388:1-22, 2019.

[13] Z. Liu and X. Li, The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing, SIAM J. Sci. Comput., 42(3):B630-B655, 2020.

[14] J. Shen, C. Wang, X. Wang, and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50:105-125, 2012.

[15] J. Shen, J. Xu, and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353:407-416, 2018.

[16] J. Shen, J. Xu, and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61:474-506, 2019.

[17] J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28(4):1669-1691, 2010.

[18] J. Shin, H. G. Lee, and J.-Y. Lee, First and second order numerical methods based on a new convex splitting for phase-field crystal equation, J. Comput. Phys., 327:519-542, 2016.

[19] N. Valizadeh and T. Rabczuk, Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach, Comput. Methods Appl. Mech. Engrg., 388:114191, 2022.

[20] X. Wang, L. Ju, and Q. Du, Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models, J. Comput. Phys., 316:21-38, 2016.

[21] X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327:294-316, 2016.

[22] X. Yang and D. Han, Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model, J. Comput. Phys., 330:1116-1134, 2017.

[23] X. Yang and H. Yu, Efficient second order unconditionally stable schemes for a phase field moving contact line model using an invariant energy quadratization approach, SIAM J. Sci. Comput., 40:B889-B914, 2018.

[24] X. Yang, J. Zhao, and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys., 333:104-127, 2017.

[25] Z. Yang and S. Dong, A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity, J. Comput. Phys., 404:109121, 2020.

[26] Z. Zhang, Y. Gong, and J. Zhao, A remark on the invariant energy quadratization (IEQ) method for preserving the original energy dissipation laws, Electron. Res. Arch., 30(2):701-714, 2022.

[27] J. Zhao, A revisit of the energy quadratization method with a relaxation technique, Appl. Math. Lett., 120:107331, 2021.