A two-derivative time integrator for the Cahn-Hilliard equation
Abstract
This paper presents a two-derivative energy-stable method for the Cahn-Hilliard equation. We use a fully implicit time discretization with the addition of two stabilization terms to maintain the energy stability. As far as we know, this is the first time an energy-stable multiderivative method has been developed for phase-field models. We present numerical results of the novel method to support our mathematical analysis. In addition, we perform numerical experiments of two multiderivative predictor-corrector methods of fourth and sixth-order accuracy, and we show numerically that all the methods are energy stable.
Keyword : multiderivative methods, high-order methods, Cahn-Hilliard equation, energy-stable methods
How to Cite
Theodosiou, E., Bringedal, C., & Schütz, J. (2024). A two-derivative time integrator for the Cahn-Hilliard equation. Mathematical Modelling and Analysis, 29(4), 714–730. https://doi.org/10.3846/mma.2024.20646
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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K. Cheng, C. Wang, S.M. Wise and X. Yue. A Second-Order, Weakly EnergyStable Pseudo-spectral Scheme for the Cahn–Hilliard Equation and its Solution by the Homogeneous Linear Iteration Method. J. Sci. Comput., 69(3):1083–1114, 2016. https://doi.org/10.1007/s10915-016-0228-3
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H. Fakih. A Cahn–Hilliard equation with a proliferation term for biological and chemical applications. Asymptotic Analysis, 94(1-2):71–104, 2015. https://doi.org/10.3233/asy-151306
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J. Guo, C. Wang, S.M. Wise and X. Yue. An H2 convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation. Communications in Mathematical Sciences, 14(2):489–515, 2016. https://doi.org/10.4310/cms.2016.v14.n2.a8
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H.L. Liao, B. Ji, L. Wang and Z. Zhang. Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn–Hilliard model. J. Sci. Comput., 92(2), 2022. https://doi.org/10.1007/s10915-022-01861-4
C. Liu, F. Frank and B.M. Rivière. Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation. Numerical Methods for Partial Differential Equations, 35(4):1509–1537, 2019. https://doi.org/10.1002/num.22362
A. Novick-Cohen. Chapter 4 the Cahn–Hilliard equation. In Handbook of Differential Equations: Evolutionary Equations, pp. 201–228. Elsevier, 2008. https://doi.org/10.1016/s1874-5717(08)00004-2
J. Schütz and D. Seal. An asymptotic preserving semi-implicit multiderivative solver. Applied Numerical Mathematics, 160:84–101, 2021. https://doi.org/10.1016/j.apnum.2020.09.004
J. Schütz, D.C. Seal and J. Zeifang. Parallel-in-time high-order multiderivative IMEX methods. J. Sci. Comput., 90(54), 2022. https://doi.org/10.1007/s10915-021-01733-3
J. Schütz, D.C. Seal and J. Zeifang. Parallel-in-time high-order multiderivative IMEX solvers. J. Sci. Comput., 90(54):1–33, 2022. https://doi.org/10.1016/j.apnum.2020.09.004
J. Schütz, D.C. Seal and A. Jaust. Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations. J. Sci. Comput., 73(2-3):1145–1163, 2017. https://doi.org/10.1007/s10915-017-0485-9
D.C. Seal, Y. Güçlü and A. Christlieb. High-order multiderivative time integrators for hyperbolic conservation laws. J. Sci. Comput., 60:101–140, 2014. https://doi.org/10.1007/s10915-013-9787-8
J. Shen and X. Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 28(4):1669– 1691, 2010. https://doi.org/10.3934/dcds.2010.28.1669
H. Song. Energy SSP-IMEX Runge–Kutta methods for the Cahn–Hilliard equation. Journal of Computational and Applied Mathematics, 292:576–590, 2016. https://doi.org/10.1016/j.cam.2015.07.030
H. Song and C.W. Shu. Unconditional Energy Stability Analysis of a Second Order Implicit–Explicit Local Discontinuous Galerkin Method for the Cahn–Hilliard Equation. J. Sci. Comput., 73(2-3):1178–1203, 2017. https://doi.org/10.1007/s10915-017-0497-5
Y. Ugurlu and D. Kaya. Solutions of the Cahn–Hilliard equation. Computers & Mathematics with Applications, 56(12):3038–3045, 2008. https://doi.org/10.1016/j.camwa.2008.07.007
F.J. Vermolen, A. Segal and A. Gefen. A pilot study of a phenomenological model of adipogenesis in maturing adipocytes using Cahn-Hilliard theory. Med. Biol. Eng. Comput., 49(12):1447–1457, 2011. https://doi.org/10.1007/s11517-011-0802-7
L. Wang and H. Yu. Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation. J. Math. Study, 51(1):89–114, 2018. https://doi.org/10.4208/jms.v51n1.18.06
L. Wang and H. Yu. On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn-Hilliard Phase-Field Equation. J. Sci. Comput., 77(2):1185–1209, 2018. https://doi.org/10.1007/s10915-018-0746-2
L. Wang and H. Yu. An energy stable linear diffusive Crank–Nicolson scheme for the Cahn–Hilliard gradient flow. Journal of Computational and Applied Mathematics, 377:112880, 2020. https://doi.org/10.1016/j.cam.2020.112880
X. Wu, G.J. van Zwieten and K.G. van der Zee. Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models. International Journal for Numerical Methods in Biomedical Engineering, 30(2):180–203, 2013. https://doi.org/10.1002/cnm.2597
Y. Yan, W. Chen, C. Wang and S.M. Wise. A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation. Communications in Computational Physics, 23(2), 2018. https://doi.org/10.4208/cicp.oa-2016-0197
J. Yang, J. Wang and J. Kim. Energy-stable method for the Cahn–Hilliard equation in arbitrary domains. International Journal of Mechanical Sciences, 228:107489, 2022. https://doi.org/10.1016/j.ijmecsci.2022.107489
J. Zeifang, A.T. Manikantan and J. Schütz. Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method. Applied Mathematics and Computation, 457:128198, 2023. https://doi.org/10.1016/j.amc.2023.128198
J. Zeifang and J. Schütz. Implicit two-derivative deferred correction time discretization for the discontinuous Galerkin method. Journal of Computational Physics, 464:111353, 2022. https://doi.org/10.1016/j.jcp.2022.111353
J. Zeifang, J. Schütz and D. Seal. Stability of implicit multiderivative deferred correction methods. BIT Numerical Mathematics, 62:1487–1503, 2022. https://doi.org/10.1007/s10543-022-00919-x
V.E. Badalassi, H.D. Ceniceros and S. Banerjee. Computation of multiphase systems with phase field models. Journal of Computational Physics, 190(2):371– 397, 2003. https://doi.org/10.1016/S0021-9991(03)00280-8
F. Bai, X. He, X. Yang, R. Zhou and C. Wang. Three dimensional phasefield investigation of droplet formation in microfluidic flow focusing devices with experimental validation. International Journal of Multiphase Flow, 93:130–141, 2017. https://doi.org/10.1016/j.ijmultiphaseflow.2017.04.008
A.R. Balakrishna and C.W. Carter. Combining phase-field crystal methods with a Cahn-Hilliard model for binary alloys. Physical Review E, 97(4):043304, 2018. https://doi.org/10.1103/PhysRevE.97.043304
M. Biskup, L. Chayes and R. Kotecký. On the formation/dissolution of equilibrium droplets. Europhysics Letters, 60(1):21–27, 2002. https://doi.org/10.1209/epl/i2002-00312-y
J.W. Cahn and J.E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28(2):258–267, 1958. https://doi.org/10.1063/1.1744102
K. Cheng, C. Wang, S.M. Wise and X. Yue. A Second-Order, Weakly EnergyStable Pseudo-spectral Scheme for the Cahn–Hilliard Equation and its Solution by the Homogeneous Linear Iteration Method. J. Sci. Comput., 69(3):1083–1114, 2016. https://doi.org/10.1007/s10915-016-0228-3
L. Cherfils, A. Miranville and S. Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 19(7):2013–2026, 2014. https://doi.org/10.3934/dcdsb.2014.19.2013
A.E. Diegel, C. Wang, X. Wang and S.M. Wise. Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system. Numer. Math., 137(3):495–534, 2017. https://doi.org/10.1007/s00211-017-0887-5
A.E. Diegel, C. Wang and S.M. Wise. Stability and Convergence of a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation. Communications in Mathematical Sciences, 14(2), 2014. Available on Internet: https://arxiv.org/pdf/1411.5248.pdf
L. Dong, C. Wang, S.M. Wise and Z. Zhang. A positivity-preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters. Journal of Computational Physics, 442:110451, 2021. https://doi.org/10.1016/j.jcp.2021.110451
C.M. Elliott and A.M. Stuart. The global dynamics of discrete semilinear parabolic equations. SIAM Journal on Numerical Analysis, 30(6):1622–1663, 1993. https://doi.org/10.1137/0730084
D.J. Eyre. Unconditionally gradient stable time marching the CahnHilliard equation. MRS Online Proceedings Library, 529:39, 1998. https://doi.org/10.1557/PROC-529-39
H. Fakih. A Cahn–Hilliard equation with a proliferation term for biological and chemical applications. Asymptotic Analysis, 94(1-2):71–104, 2015. https://doi.org/10.3233/asy-151306
V.L. Ginzburg. On the theory of superconductivity. Il Nuovo Cimento (19551965), 2:1234–1250, 1955. https://doi.org/10.1007/BF02731579
J. Guo, C. Wang, S.M. Wise and X. Yue. An H2 convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation. Communications in Mathematical Sciences, 14(2):489–515, 2016. https://doi.org/10.4310/cms.2016.v14.n2.a8
R. Guo and Y. Xu. Efficient solvers of discontinuous Galerkin discretization for the Cahn–Hilliard equations. J. Sci. Comput., 58(2):380–408, 2013. https://doi.org/10.1007/s10915-013-9738-4
E. Hairer and G. Wanner. Multistep-multistage-multiderivative methods for ordinary differential equations. Computing, 11(3):287–303, 1973. https://doi.org/10.1007/BF02252917
A. Jaust, J. Schütz and D.C. Seal. Implicit multistage two-derivative discontinuous Galerkin schemes for viscous conservation laws. J. Sci. Comput., 69:866–891, 2016. https://doi.org/10.1007/s10915-016-0221-x
J. Kim. Phase-Field Models for Multi-Component Fluid Flows. Communications in Computational Physics, 12(3):613–661, 2012. https://doi.org/10.4208/cicp.301110.040811a
J. Kim, S. Lee, Y. Choi, S.M. Lee and D. Jeong. Basic principles and practical applications of the Cahn–Hilliard equation. Mathematical Problems in Engineering, 2016:1–11, 2016. https://doi.org/10.1155/2016/9532608
H.L. Liao, B. Ji, L. Wang and Z. Zhang. Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn–Hilliard model. J. Sci. Comput., 92(2), 2022. https://doi.org/10.1007/s10915-022-01861-4
C. Liu, F. Frank and B.M. Rivière. Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation. Numerical Methods for Partial Differential Equations, 35(4):1509–1537, 2019. https://doi.org/10.1002/num.22362
A. Novick-Cohen. Chapter 4 the Cahn–Hilliard equation. In Handbook of Differential Equations: Evolutionary Equations, pp. 201–228. Elsevier, 2008. https://doi.org/10.1016/s1874-5717(08)00004-2
J. Schütz and D. Seal. An asymptotic preserving semi-implicit multiderivative solver. Applied Numerical Mathematics, 160:84–101, 2021. https://doi.org/10.1016/j.apnum.2020.09.004
J. Schütz, D.C. Seal and J. Zeifang. Parallel-in-time high-order multiderivative IMEX methods. J. Sci. Comput., 90(54), 2022. https://doi.org/10.1007/s10915-021-01733-3
J. Schütz, D.C. Seal and J. Zeifang. Parallel-in-time high-order multiderivative IMEX solvers. J. Sci. Comput., 90(54):1–33, 2022. https://doi.org/10.1016/j.apnum.2020.09.004
J. Schütz, D.C. Seal and A. Jaust. Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations. J. Sci. Comput., 73(2-3):1145–1163, 2017. https://doi.org/10.1007/s10915-017-0485-9
D.C. Seal, Y. Güçlü and A. Christlieb. High-order multiderivative time integrators for hyperbolic conservation laws. J. Sci. Comput., 60:101–140, 2014. https://doi.org/10.1007/s10915-013-9787-8
J. Shen and X. Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 28(4):1669– 1691, 2010. https://doi.org/10.3934/dcds.2010.28.1669
H. Song. Energy SSP-IMEX Runge–Kutta methods for the Cahn–Hilliard equation. Journal of Computational and Applied Mathematics, 292:576–590, 2016. https://doi.org/10.1016/j.cam.2015.07.030
H. Song and C.W. Shu. Unconditional Energy Stability Analysis of a Second Order Implicit–Explicit Local Discontinuous Galerkin Method for the Cahn–Hilliard Equation. J. Sci. Comput., 73(2-3):1178–1203, 2017. https://doi.org/10.1007/s10915-017-0497-5
Y. Ugurlu and D. Kaya. Solutions of the Cahn–Hilliard equation. Computers & Mathematics with Applications, 56(12):3038–3045, 2008. https://doi.org/10.1016/j.camwa.2008.07.007
F.J. Vermolen, A. Segal and A. Gefen. A pilot study of a phenomenological model of adipogenesis in maturing adipocytes using Cahn-Hilliard theory. Med. Biol. Eng. Comput., 49(12):1447–1457, 2011. https://doi.org/10.1007/s11517-011-0802-7
L. Wang and H. Yu. Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation. J. Math. Study, 51(1):89–114, 2018. https://doi.org/10.4208/jms.v51n1.18.06
L. Wang and H. Yu. On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn-Hilliard Phase-Field Equation. J. Sci. Comput., 77(2):1185–1209, 2018. https://doi.org/10.1007/s10915-018-0746-2
L. Wang and H. Yu. An energy stable linear diffusive Crank–Nicolson scheme for the Cahn–Hilliard gradient flow. Journal of Computational and Applied Mathematics, 377:112880, 2020. https://doi.org/10.1016/j.cam.2020.112880
X. Wu, G.J. van Zwieten and K.G. van der Zee. Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models. International Journal for Numerical Methods in Biomedical Engineering, 30(2):180–203, 2013. https://doi.org/10.1002/cnm.2597
Y. Yan, W. Chen, C. Wang and S.M. Wise. A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation. Communications in Computational Physics, 23(2), 2018. https://doi.org/10.4208/cicp.oa-2016-0197
J. Yang, J. Wang and J. Kim. Energy-stable method for the Cahn–Hilliard equation in arbitrary domains. International Journal of Mechanical Sciences, 228:107489, 2022. https://doi.org/10.1016/j.ijmecsci.2022.107489
J. Zeifang, A.T. Manikantan and J. Schütz. Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method. Applied Mathematics and Computation, 457:128198, 2023. https://doi.org/10.1016/j.amc.2023.128198
J. Zeifang and J. Schütz. Implicit two-derivative deferred correction time discretization for the discontinuous Galerkin method. Journal of Computational Physics, 464:111353, 2022. https://doi.org/10.1016/j.jcp.2022.111353
J. Zeifang, J. Schütz and D. Seal. Stability of implicit multiderivative deferred correction methods. BIT Numerical Mathematics, 62:1487–1503, 2022. https://doi.org/10.1007/s10543-022-00919-x