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Journal of Scientific Computing

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05-04-2019

Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints

Authors: Xiaobo Jing, Jun Li, Xueping Zhao, Qi Wang

Published in: Journal of Scientific Computing | Issue 1/2019

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Abstract

We present a set of linear, second order, unconditionally energy stable schemes for the Allen-Cahn equation with nonlocal constraints that preserves the total volume of each phase in a binary material system. The energy quadratization strategy is employed to derive the energy stable semi-discrete numerical algorithms in time. Solvability conditions are then established for the linear systems resulting from the semi-discrete, linear schemes. The fully discrete schemes are obtained afterwards by applying second order finite difference methods on cell-centered grids in space. The performance of the schemes are assessed against two benchmark numerical examples, in which dynamics obtained using the volume-preserving Allen-Cahn equations with nonlocal constraints is compared with those obtained using the classical Allen-Cahn as well as the Cahn-Hilliard model, respectively, demonstrating slower dynamics when volume constraints are imposed as well as their usefulness as alternatives to the Cahn–Hilliard equation in describing phase evolutionary dynamics for immiscible material systems while preserving the phase volumes. Some performance enhancing, practical implementation methods for the linear energy stable schemes are discussed in the end.

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Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica 27(6), 1085–1095 (1979)CrossRef

Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. i. interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)CrossRef

Gurtin, M.E., Polignone, D., Vinals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(06), 815–831 (1996)MathSciNetMATHCrossRef

Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics, vol. 73. Oxford University Press, Oxford (1988)

Leslie, F.M.: Theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1–81 (1979)CrossRef

Gong, Y., Zhao, J., Yang, X., Wang, Q.: Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities. SIAM J. Sci. Comput. 40(1), B138–B167 (2018)MathSciNetMATHCrossRef

Pethrick, R.A.: The theory of polymer dynamics m. doi and s. f. edwards, oxford university press. J. Chem. Technol. Biotechnol. 44(1), 79–80 (1988)

Gong, Y., Zhao, J., Wang, Q.: Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation. Adv. Comput. Math. 44(5), 1573–1600 (2018)MathSciNetMATHCrossRef

Yang, X., Lili, J.: Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Methods Appl. Mech. Eng. 318, 1005–1029 (2017)MathSciNetCrossRef

10.

Zhao, J., Wang, Q., Yang, X.: Numerical approximations to a new phase field model for two phase flows of complex fluids. Comput. Methods Appl. Mech. Eng. 310, 77–97 (2016)MathSciNetCrossRef

11.

Zhao, J., Wang, Q., Yang, X.: Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. Int. J. Numer. Methods Eng. 110(3), 279–300 (2017)MathSciNetMATHCrossRef

12.

Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47(3), 2269–2288 (2009)MathSciNetMATHCrossRef

13.

Yang, X., Gong, Y., Li, J., Zhao, J., Wang, Q.: On hydrodynamic phase field models for binary fluid mixtures. Theor. Comput. Fluid Dynam. 32(5), 1–24 (2017)MathSciNet

14.

Cheng, Y., Kurganov, A., Zhuolin, Q., Tang, T.: Fast and stable explicit operator splitting methods for phase-field models. J. Comput. Phys. 303, 45–65 (2015)MathSciNetMATHCrossRef

15.

Guo, R., Yan, X.: Local discontinuous galerkin method and high order semi-implicit scheme for the phase field crystal equation. SIAM J. Sci. Comput. 38(1), A105–A127 (2016)MathSciNetMATHCrossRef

16.

Qiang, D., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198(2), 450–468 (2004)MathSciNetMATHCrossRef

17.

Li, H., Lili, J., Zhang, C., Peng, Q.: Unconditionally energy stable linear schemes for the diffuse interface model with peng-robinson equation of state. J. Sci. Comput. 75(2), 993–1015 (2018)MathSciNetMATHCrossRef

18.

Rubinstein, J., Sternberg, P.: Nonlocal reaction diffusion equations and nucleation. IMA J. Appl. Math. 48(3), 249–264 (1992)MathSciNetMATHCrossRef

19.

Yang, X., Feng, J.J., Liu, C., Shen, J.: Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys. 218(1), 417–428 (2006)MathSciNetMATHCrossRef

20.

Mellenthin, J., Karma, A., Plapp, M.: Phase-field crystal study of grain-boundary premelting. Phys. Rev. B 78(18), 184110 (2008)CrossRef

21.

Guillén-González, F., Tierra, G.: On linear schemes for a cahn-hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013)MathSciNetMATHCrossRef

22.

Li, D., Qiao, Z.: On second order semi-implicit fourier spectral methods for 2d cahn-hilliard equations. J. Sci. Comput. 70(1), 301–341 (2017)MathSciNetMATHCrossRef

23.

Elliott, C.M., Stuart, A.M.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30(6), 1622–1663 (1993)MathSciNetMATHCrossRef

24.

Fan, X., Kou, J., Qiao, Z., Sun, S.: A componentwise convex splitting scheme for diffuse interface models with van der waals and peng-robinson equations of state. SIAM J. Sci. Comput. 39(1), B1–B28 (2017)MathSciNetMATHCrossRef

25.

Eyre, D.J.: Unconditionally gradient stable time marching the cahn-hilliard equation. MRS online proceedings library archive, 529 (1998)

26.

Shen, J., Wang, C., Wang, X., Wise, S.M.: Second-order convex splitting schemes for gradient flows with ehrlich-schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50(1), 105–125 (2012)MathSciNetMATHCrossRef

27.

Wang, C., Wang, X., Wise, S.M.: Unconditionally stable schemes for equations of thin film epitaxy. Discrete Contin. Dyn. Syst. 28(1), 405–423 (2010)MathSciNetMATHCrossRef

28.

Han, D., Wang, X.: A second order in time, uniquely solvable, unconditionally stable numerical scheme for cahn-hilliard-navier-stokes equation. J. Comput. Phys. 290, 139–156 (2015)MathSciNetMATHCrossRef

29.

Chuanju, X., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44(4), 1759–1779 (2006)MathSciNetMATHCrossRef

30.

Shen, J., Yang, X.: Numerical approximations of allen-cahn and cahn-hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010)MathSciNetMATHCrossRef

31.

Chen, W., Han, D., Wang, X.: Uniquely solvable and energy stable decoupled numerical schemes for the cahn-hilliard-stokes-darcy system for two-phase flows in karstic geometry. Numer. Math. 137(1), 229–255 (2017)MathSciNetMATHCrossRef

32.

Wang, L., Yu, H.: On efficient second order stabilized semi-implicit schemes for the cahn-hilliard phase-field equation. J. Sci. Comput. 77(2), 1–25 (2017)MathSciNet

33.

Qiang, D., Lili, J., Li, X., Qiao, Z.: Stabilized linear semi-implicit schemes for the nonlocal cahn-hilliard equation. J. Comput. Phys. 363, 39–54 (2018)MathSciNetMATHCrossRef

34.

Wang, L., Yu, H.: Energy stable second order linear schemes for the allen-cahn phase-field equation. arXiv preprint arXiv:1807.03171 (2018)

35.

Zhao, J., Yang, X., Gong, Y., Zhao, X., Yang, X., Li, J., Wang, Q.: A general strategy for numerical approximations of non-equilibrium models-part i: thermodynamical systems. Int. J. Numer. Anal. Model. 15(6), 884–918 (2018)MathSciNetMATH

36.

Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104–127 (2017)MathSciNetMATHCrossRef

37.

Shen, J., Jie, X., Yang, J.: The scalar auxiliary variable (sav) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)MathSciNetMATHCrossRef

38.

Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. arXiv preprint arXiv:1710.01331, (2017)

39.

Li, X., Shen, J., Rui, H.: Energy stability and convergence of sav block-centered finite difference method for gradient flows. arXiv preprint arXiv:1812.01793 (2018)

40.

Dong, S., Yang, Z., Lin, L.: A family of second-order energy-stable schemes for cahn-hilliard type equations. arXiv preprint arXiv:1803.06047 (2018)

41.

Zhao, J., Gong, Y., Wang, Q.: Aritrary high order unconditionally energy stable schemes for gradient flow models. Submitted to J. Comput. Phys. (2019)

42.

Chen, L., Zhao, J., Yang, X.: Regularized linear schemes for the molecular beam epitaxy model with slope selection. Appl. Numer. Math. 128, 139–156 (2018)MathSciNetMATHCrossRef

43.

Yang, X.: Efficient schemes with unconditionally energy stability for the anisotropic cahn-hilliard equation using the stabilized-scalar augmented variable (s-sav) approach. arXiv preprint arXiv:1804.02619 (2018)

44.

Zhao, J., Yang, X., Li, J., Wang, Q.: Energy stable numerical schemes for a hydrodynamic model of nematic liquid crystals. SIAM J. Sci. Comput. 38(5), A3264–A3290 (2016)MathSciNetMATHCrossRef

45.

Zhao, J., Yang, X., Gong, Y., Wang, Q.: A novel linear second order unconditionally energy stable scheme for a hydrodynamic-tensor model of liquid crystals. Comput. Methods Appl. Mech. Eng. 318, 803–825 (2017)MathSciNetCrossRef

46.

Zhao, J., Yang, X., Shen, J., Wang, Q.: A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids. J. Comput. Phys. 305, 539–556 (2016)MathSciNetMATHCrossRef

47.

Onsager, L.: Reciprocal relations in irreversible processes. i. Phys. Rev. 37(4), 405 (1931)MATHCrossRef

48.

Onsager, L.: Reciprocal relations in irreversible processes. ii. Phys. Rev. 38(12), 2265 (1931)MATHCrossRef

49.

Qian, T., Wang, X.-P., Sheng, P.: A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360 (2006)MathSciNetMATHCrossRef

50.

Xianmin, X., Di, Y., Haijun, Y.: Sharp-interface limits of a phase-field model with a generalized navier slip boundary condition for moving contact lines. J. Fluid Mech. 849, 805–833 (2018)MathSciNetMATHCrossRef

51.

Sun, Shouwen, Jing, Xiaobo, Wang, Qi: Error estimates of energy stable numerical schemes for allen–cahn equations with nonlocal constraints. Journal of Scientific Computing, pp. 1–31 (2018)

52.

Bodewig, E.: Matrix calculus, north, p17 (1959)

Title: Second Order Linear Energy Stable Schemes for Allen-Cahn Equations with Nonlocal Constraints
Authors: Xiaobo Jing
Jun Li
Xueping Zhao
Qi Wang
Publication date: 05-04-2019
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-00946-x

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