Abstract
The conservative Allen–Cahn equation with a nonlocal Lagrange multiplier satisfies mass conservation and energy dissipation property. A challenge to numerically solving the equation is how to treat the nonlinear and nonlocal terms to preserve mass conservation and energy stability without compromising accuracy. To resolve this problem, we first apply the convex splitting idea to not only the term corresponding to the Allen–Cahn equation but also the nonlocal term. A wise implementation of the convex splitting for the nonlocal term ensures numerically exact mass conservation. And we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method. We show analytically that the scheme is uniquely solvable and unconditionally energy stable by using the fact that the scheme guarantees exact mass conservation. Numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme.
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References
Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)
Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)
Baskaran, A., Hu, Z., Lowengrub, J.S., Wang, C., Wise, S.M., Zhou, P.: Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. J. Comput. Phys. 250, 270–292 (2013)
Beneš, M., Yazaki, S., Kimura, M.: Computational studies of non-local anisotropic Allen–Cahn equation. Math. Bohem. 136, 429–437 (2011)
Brassel, M., Bretin, E.: A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Meth. Appl. Sci. 34, 1157–1180 (2011)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
Chen, L.-Q.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002)
Chen, W., Wang, C., Wang, S., Wang, X., Wise, S.M.: Energy stable numerical schemes for ternary Cahn–Hilliard system. J. Sci. Comput. 84, 27 (2020)
Elliott, C.M., Stuart, A.M.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30, 1622–1663 (1993)
Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. MRS Proc. 529, 39–46 (1998)
Hong, Q., Gong, Y., Zhao, J., Wang, Q.: Arbitrarily high order structure-preserving algorithms for the Allen–Cahn model with a nonlocal constraint. Appl. Numer. Math. 170, 321–339 (2021)
Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S.: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J. Comput. Phys. 228, 5323–5339 (2009)
Huang, Z., Lin, G., Ardekani, A.M.: Consistent and conservative scheme for incompressible two-phase flows using the conservative Allen–Cahn model. J. Comput. Phys. 420, 109718 (2020)
Jeong, D., Kim, J.: Conservative Allen–Cahn–Navier–Stokes system for incompressible two-phase fluid flows. Comput. Fluids 156, 239–246 (2017)
Jing, X., Li, J., Zhao, X., Wang, Q.: Second order linear energy stable schemes for Allen–Cahn equations with nonlocal constraints. J. Sci. Comput. 80, 500–537 (2019)
Kim, J., Lee, S., Choi, Y.: A conservative Allen–Cahn equation with a space-time dependent Lagrange multiplier. Int. J. Engrg. Sci. 84, 11–17 (2014)
Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44, 139–181 (2003)
Lee, D.: The numerical solutions for the energy-dissipative and mass-conservative Allen–Cahn equation. Comput. Math. Appl. 80, 263–284 (2020)
Lee, H.G., Shin, J., Lee, J.-Y.: First- and second-order energy stable methods for the modified phase field crystal equation. Comput. Methods Appl. Mech. Engrg. 321, 1–17 (2017)
Okumura, M.: A stable and structure-preserving scheme for a non-local Allen–Cahn equation. Japan J. Indust. Appl. Math. 35, 1245–1281 (2018)
Pareschi, L., Russo, G.: Implicit-explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)
Rubinstein, J., Sternberg, P.: Nonlocal reaction–diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264 (1992)
Shen, J., Yang, X.: An efficient moving mesh spectral method for the phase-field model of two-phase flows. J. Comput. Phys. 228, 2978–2992 (2009)
Shen, J., Yang, X.: A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput. 32, 1159–1179 (2010)
Shin, J., Lee, H.G., Lee, J.-Y.: First and second order numerical methods based on a new convex splitting for phase-field crystal equation. J. Comput. Phys. 327, 519–542 (2016)
Shin, J., Lee, H.G., Lee, J.-Y.: Unconditionally stable methods for gradient flow using Convex Splitting Runge–Kutta scheme. J. Comput. Phys. 347, 367–381 (2017)
Sun, S., Jing, X., Wang, Q.: Error estimates of energy stable numerical schemes for Allen–Cahn equations with nonlocal constraints. J. Sci. Comput. 79, 593–623 (2019)
Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977)
Yang, X.: A novel fully decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier–Stokes equations coupled with mass-conserved Allen–Cahn phase-field model of two-phase incompressible flow. Int. J. Numer. Methods Eng. 122, 1283–1306 (2021)
Yuan, M., Chen, W., Wang, C., Wise, S.M., Zhang, Z.: An energy stable finite element scheme for the three-component Cahn–Hilliard-type model for macromolecular microsphere composite hydrogels. J. Sci. Comput. 87, 78 (2021)
Zhai, S., Weng, Z., Feng, X.: Investigations on several numerical methods for the non-local Allen–Cahn equation. Int. J. Heat Mass Transfer 87, 111–118 (2015)
Zhang, Z., Tang, H.: An adaptive phase field method for the mixture of two incompressible fluids. Comput. Fluids 36, 1307–1318 (2007)
Acknowledgements
The authors thank the reviewers for the constructive and helpful comments on the revision of this article. This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT of Korea (MSIT) (Nos. 2019R1A6A1A11051177, 2019R1C1C1011112, 2020R1C1C1A01013468).
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Lee, H.G., Shin, J. & Lee, JY. A High-Order and Unconditionally Energy Stable Scheme for the Conservative Allen–Cahn Equation with a Nonlocal Lagrange Multiplier. J Sci Comput 90, 51 (2022). https://doi.org/10.1007/s10915-021-01735-1
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DOI: https://doi.org/10.1007/s10915-021-01735-1
Keywords
- Conservative Allen–Cahn equation
- Convex splitting
- Mass conservation
- Unconditional unique solvability
- Unconditional energy stability
- High-order time accuracy