Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation
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Seunggyu Lee
Abstract
We propose a fourth-order spatial and second-order temporal accurate and unconditionally stable compact finite-difference scheme for the Cahn–Hilliard equation. The proposed scheme has a higher-order accuracy in space than conventional central difference schemes even though both methods use a three-point stencil. Its compactness may be useful when applying the scheme to numerical implementation. In a temporal discretization, the secant-type algorithm, which is known as the second-order accurate scheme, is applied. Furthermore, the unique solvability regardless of the temporal and spatial step size, unconditionally gradient stability, and discrete mass conservation are proven. It guarantees that large temporal and spatial step sizes could be used with the high-order accuracy and the original properties of the CH equation. Then, numerical results are presented to confirm the efficiency and accuracy of the proposed scheme. The efficiency of the proposed scheme is better than other low order accurate stable schemes.
Acknowledgements:
The author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B1001937) and the National Institute for Mathematical Sciences (NIMS) grant funded by the Korean government.
References
[1] J. Cahn, J. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys. 28 (2) (1958), 258–267.10.1063/1.1744102Suche in Google Scholar
[2] D. Lee, J.-Y. Huh, D. Jeong, J. Shin, A. Yun, J. Kim, Physical, mathematical, and numerical derivations of the Cahn–Hilliard equations, Comput. Mater. Sci. 81 (2014), 216–255.10.1016/j.commatsci.2013.08.027Suche in Google Scholar
[3] A. Bertozzi, S. Esedoglu, A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE T. Image Process. 16 (1) (2007), 285–291.10.1109/TIP.2006.887728Suche in Google Scholar
[4] M. Burger, L. He, C. Schönlieb, Cahn–Hilliard inpainting and a generalization of grayvalue images, SIAM J. Imaging Sci. 2 (4) (2009), 1129–1167.10.1137/080728548Suche in Google Scholar
[5] E. Khain, L. Sander, Generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E. 77 (2008), 051129.10.1103/PhysRevE.77.051129Suche in Google Scholar
[6] M. Burger, R. Stainko, Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim. 45 (4) (2006), 1447–1466.10.1137/05062723XSuche in Google Scholar
[7] S. Zhou, M. Wang, Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition, Struct. Multidiscip. O. 33 (2) (2007), 89–111.10.1007/s00158-006-0035-9Suche in Google Scholar
[8] D. Jeong, S. Lee, Y. Choi, J. Kim, Energy-minimizing wavelengths of equilibrium states for diblock copolymers in the hex-cylinder phase, Curr. Appl. Phys. 15 (2015), 799–804.10.1016/j.cap.2015.04.033Suche in Google Scholar
[9] D. Jeong, J. Shin, Y. Li, Y. Choi, J.-H. Jung, S. Lee, J. Kim, Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers, Curr. Appl. Phys. 14 (2014), 1263–1272.10.1016/j.cap.2014.06.016Suche in Google Scholar
[10] S. Hu, L. Chen, A phase-field model for evolving microstructures with strong inhomogeneity, Acta Mater. 49 (11) (2001), 1879–1890.10.1016/S1359-6454(01)00118-5Suche in Google Scholar
[11] D. Jeong, S. Lee, J. Kim, An efficient numerical method for evolving microstructures with strong elastic inhomogeneity, Model. Simul. Mater. Sci. Eng. 23 (2015), 045007.10.1088/0965-0393/23/4/045007Suche in Google Scholar
[12] S. Lee, Y. Choi, D. Lee, H.-K. Jo, S. Lee, S, Myung, J. Kim, A modified Cahn–Hilliard equation for 3d volume reconstruction from two planar cross sections, J. KSIAM. 19 (1) (2015), 47–56.10.12941/jksiam.2015.19.047Suche in Google Scholar
[13] Y. Li, J. Shin, Y. Choi, J. Kim, Three-dimensional volume reconstruction from slice data using phase-field models, Comput. Vis. Image Und. 137 (2015), 115–124.10.1016/j.cviu.2015.02.001Suche in Google Scholar
[14] V. Cristini, X. Li, J. Lowengrub, S. Wise, Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching, J. Math. Biol. 58 (2009), 723–763.10.1007/s00285-008-0215-xSuche in Google Scholar PubMed PubMed Central
[15] X.Wu, G. Zwieten, K. Zee, Stabilized second-order convex splitting schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models, Int. J. Numer. M. Biomed. Eng. 30 (2) (2014), 180–203.10.1002/cnm.2597Suche in Google Scholar PubMed
[16] C. Elliott, D. French, Numerical studies of the Cahn–Hilliard equation for phase separation, IMA J. Appl. Math. 38 (2) (1987), 97–128.10.1093/imamat/38.2.97Suche in Google Scholar
[17] D. Eyer, Unconditionally gradient stable scheme marching the Cahn–Hilliard equation, MRS Proceedings 529 (1998), 39–46.10.1557/PROC-529-39Suche in Google Scholar
[18] Y. He, Y. Liu, T. Tang, On large time-stepping methods for the Cahn–Hilliard equation, Appl. Numer. Math. 57 (5) (2007), 616–628.10.1016/j.apnum.2006.07.026Suche in Google Scholar
[19] C. Lee, D. Jeong, J. Shin, Y. Li, J. Kim, A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation, Physica A 409 (2014), 17–28.10.1016/j.physa.2014.04.038Suche in Google Scholar
[20] S. Lee, C. Lee, H. Lee, J. Kim, Comparison of different numerical schemes for the Cahn–Hilliard equation, J. KSIAM. 17 (3) (2013), 197–207.10.12941/jksiam.2013.17.197Suche in Google Scholar
[21] Y. Li, H. Lee, B. Xia, J. Kim, A compact forth-order finite difference scheme for the three-dimensional Cahn–Hilliard eqatuion, Comput. Phys. Commun. 200 (2016), 108–116.10.1016/j.cpc.2015.11.006Suche in Google Scholar
[22] E. de Mello, O. da Silveira Filho, Numerical study of the Cahn–Hilliard equation in one, two, and three dimensions, Physica A 347 (2005), 429–443.10.1016/j.physa.2004.08.076Suche in Google Scholar
[23] T. Rogers, R. Deasi, Numerical study of late-stage coarsening for off-critical quenches in the Cahn–Hilliard equation of phase separation, Phys. Rev. B. 39 (16) (1989), 11956.10.1103/PhysRevB.39.11956Suche in Google Scholar
[24] J. Stephenson, Single cell discretizations of order two and four for biharmonic problems, J. Comput. Phys. 55 (1) (1984), 65–80.10.1016/0021-9991(84)90015-9Suche in Google Scholar
[25] Z. Tian, S. Dai, High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys. 220 (2) (2007), 952–974.10.1016/j.jcp.2006.06.001Suche in Google Scholar
[26] J. Zhang, Multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization, J. Comput. Phys. 179 (1) (2002), 170–179.10.1006/jcph.2002.7049Suche in Google Scholar
[27] M. Li, T. Tang, A compact fourth-order finite difference scheme for unsteady viscous incompressible flows, J. Sci. Comput. 16 (1) (2001), 29–45.Suche in Google Scholar
[28] E. Turkel, D. Gordon, R. Gorgon, S. Tsynkov, Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number, J. Comput. Phys. 232 (1) (2013), 272–287.10.1016/j.jcp.2012.08.016Suche in Google Scholar
[29] J. Li, Z. Sun, X. Zhao, A three level linearized compact difference scheme for the Cahn–Hilliard equation, Sci. China Math. 55 (4) (2012), 805–826.10.1007/s11425-011-4290-xSuche in Google Scholar
[30] Q. Du, R. Nicolaides, Numerical analysis of a continuum model of a phase transition. SIAM Numer. Anal. 28 (1991), 1310–1322.10.1137/0728069Suche in Google Scholar
[31] K. Atkinson, An introduction to numerical analysis, 2nd Edition, John Wiley & Sons, New York, 1988.Suche in Google Scholar
[32] L. Cherfils, A. Miranville, S. Zelik, The Cahn–Hilliard equation with logarithmic potentials, Milan J. Math. 79 (2011), 561–596.10.1007/s00032-011-0165-4Suche in Google Scholar
[33] J. Greer, A. Bertozzi, G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys. 216 (2006), 216–246.10.1016/j.jcp.2005.11.031Suche in Google Scholar
[34] U. Trottenberg, C. Oosterlee, A. Schüller, Multigrid, Academic Press, London, 2001.Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Directed Transport in Symmetrically Periodic Potentials Induced by Cross-Correlation among Colored Gaussian Noises
- Effect of Fractional Damping in Double-Well Duffing–Vander Pol Oscillator Driven by Different Sinusoidal Forces
- Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization
- Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation
- Unit Root Testing in the Presence of Mean Reverting Jumps: Evidence from US T-Bond Yields
- Thermal Analysis of Longitudinal Fin with Temperature-Dependent Properties and Internal heat Generation by a Novel Intelligent Computational Approach Using Optimized Chebyshev Polynomials
- A Stream/Block Combination Image Encryption Algorithm Using Logistic Matrix to Scramble
- Dynamic Analysis of a Composite Structure under Random Excitation Based on the Spectral Element Method
- Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations
- Representation of Solutions and Finite Time Stability for Delay Differential Systems with Impulsive Effects
- Numerical Study of the Dynamics of Particles Motion with Different Sizes from Coal-Based Thermal Power Plant
Artikel in diesem Heft
- Frontmatter
- Original Research Articles
- Directed Transport in Symmetrically Periodic Potentials Induced by Cross-Correlation among Colored Gaussian Noises
- Effect of Fractional Damping in Double-Well Duffing–Vander Pol Oscillator Driven by Different Sinusoidal Forces
- Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization
- Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation
- Unit Root Testing in the Presence of Mean Reverting Jumps: Evidence from US T-Bond Yields
- Thermal Analysis of Longitudinal Fin with Temperature-Dependent Properties and Internal heat Generation by a Novel Intelligent Computational Approach Using Optimized Chebyshev Polynomials
- A Stream/Block Combination Image Encryption Algorithm Using Logistic Matrix to Scramble
- Dynamic Analysis of a Composite Structure under Random Excitation Based on the Spectral Element Method
- Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations
- Representation of Solutions and Finite Time Stability for Delay Differential Systems with Impulsive Effects
- Numerical Study of the Dynamics of Particles Motion with Different Sizes from Coal-Based Thermal Power Plant
