Novel mass-conserving Allen-Cahn equation for the boundedness of an order parameter. (English) Zbl 1450.35133
Summary: There are several theoretically well-posed models for the Allen-Cahn equation under mass conservation. The conservative property is a gift from the additional nonlocal term playing a role of a Lagrange multiplier. However, the same term destroys the boundedness property that the original Allen-Cahn equation presents: The solution is bounded by 1 with an initial datum bounded by 1. In this paper, we propose a novel mass-conserving Allen-Cahn equation and prove the existence and uniqueness of a classical solution in the context of the theory of analytic semigroups as well as the boundedness property of the solution. From the numerical point of view, we investigate a linear unconditionally energy stable splitting scheme of the proposed model for the boundedness of numerical solutions. Various numerical experiments are presented to demonstrate the validity of the proposed method and to make distinctions from a few closely related methods.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K55 | Nonlinear parabolic equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
References:
| [1] | 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) |
| [2] | Rubinstein, J.; Sternberg, P., Nonlocal reaction-diffusion equations and nucleation, IMA J Appl Math, 48, 249-264 (1992) · Zbl 0763.35051 |
| [3] | Lee, H. G., High-order and mass conservative methods for the conservative Allen-Cahn equation, Comput Math Appl, 72, 620-631 (2016) · Zbl 1359.65216 |
| [4] | Dolcetta, I. C.; Vita, S. F.; March, R., Area preserving curve shortening flows: from phase separation to image processing, Interfaces Free Bound, 4, 325-343 (2002) · Zbl 1021.35129 |
| [5] | Blank, L.; Butz, M.; Garcke, H.; Sarbu, L.; Styles, V., Allen-Cahn and Cahn-Hilliard variational inequalities solved with optimization techniques, (Leugering, G.; Engell, S.; Hinze, M.; Rannacher, R.; Schulz, V.; Ulbrich, M.; Ulbrich, S., Constrained Optimization and Optimal Control for Partial Differential Equations, 160; (2002), ISNM), 21-35 · Zbl 1356.49009 |
| [6] | Mu, X.; Frank, F.; Riviere, B.; Alpak, F. O.; Chapman, W. G., Mass-conserved density gradient theory model for nucleation process, Ind Eng Chem Res, 57, 16476-16485 (2018) |
| [7] | Miranville, A., The Cahn-Hilliard equation and some of its variants, AIMS Math, 2, 479-544 (2017) · Zbl 1425.35086 |
| [8] | Li, Y.; Choi, J.-I.; Kim, J., A phase-field fluid modeling and computation with interfacial profile correction term, Commun Nonlinear Sci Numer Simul, 30, 84-100 (2016) · Zbl 1461.76512 |
| [9] | Lowengrub, J.; Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc R Soc London Ser A, 454, 2617-2654 (1998) · Zbl 0927.76007 |
| [10] | Bertozzi, A.; Esedoglu, S.; Gillette, A., The inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans Imag Proc, 16, 285-291 (2007) · Zbl 1279.94008 |
| [11] | Jeong, D.; Li, Y.; Choi, Y.; Yoo, M.; Kang, D.; Park, J.; Choi, J.; Kim, J., Numerical simulation of the zebra pattern formation on a three-dimensional model, Physica A, 475, 106-116 (2017) · Zbl 1400.35161 |
| [12] | Jeong, D.; Kim, J., Conservative Allen-Cahn-Navier-Stokes system for incompressible two-phase fluid flows, Comput Fluids, 156, 239-246 (2017) · Zbl 1390.76577 |
| [13] | Lee, H.; J-Y, L., A second order operator splitting method for Allen-Cahn type equations with nonlinear source terms, Physica A, 432, 24-34 (2015) · Zbl 1400.82256 |
| [14] | Garcke, H.; Nestler, B.; Stinner, B.; Wendler, F., Allen-Cahn systems with volume constraints, Math Models Methods Appl Sci, 18, 1347-1381 (2008) · Zbl 1147.49036 |
| [15] | Alfaro, M.; Hilhorst, D.; Matano, H., The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J Differ Equs, 245, 505-565 (2008) · Zbl 1154.35006 |
| [16] | Barles, G.; Bronsard, L.; Souganidis, P. E., Front propagation for reaction-diffusion equations of bistable type, Anal Nonlin, 9, 479-506 (1992) · Zbl 0794.35076 |
| [17] | Evans, L. C.; Soner, H. M.; Souganidis, P. E., Phase transitions and generalized motion by mean curvature, Commun Pure Appl Math, 45, 1097-1123 (1992) · Zbl 0801.35045 |
| [18] | Tang, T.; Yang, J., Implicit-explicit scheme for the Allen-Cahn equation preserves the maximum principle, J Comput Math, 34, 471-481 (2016) |
| [19] | Shen, J.; Tang, T.; Yang, J., On the maximum principle preserving schemes for the generalized Allen-Cahn equation, Commun Math Sci, 14, 1517-1534 (2016) · Zbl 1361.65059 |
| [20] | Bronsard, L.; Stoth, B., Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J Math Anal, 28, 769-807 (1997) · Zbl 0874.35009 |
| [21] | Chen, X.; Hilhorst, D.; Logak, E., Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interface Free Bound, 12, 527-549 (2010) · Zbl 1219.35018 |
| [22] | Ward, M. J., Metastable bubble solutions for the Allen-Cahnequation with mass conservation, SIAM J Appl Math, 56, 1247-1279 (1996) · Zbl 0870.35011 |
| [23] | Stafford, D.; Ward, M. J.; Wetton, B., The dynamics of drops and attached interfaces for the constrained Allen-Cahn equation, Eur J Appl Math, 12, 1-24 (2001) · Zbl 0988.76095 |
| [24] | Colli, P.; Fukao, T., The Allen-Cahn equation with dynamic boundary conditions and mass constraints, Math Methods Appl Sci, 38, 3950-3967 (2015) · Zbl 1334.35165 |
| [25] | Farshbaf-Shaker, M. H.; Fukao, T.; Yamazaki, N., Lagrange multiplier and singular limit of double obstacle problems for the Allen-Cahn equation with constraint, Math Meth Appl Sci, 40, 5-21 (2017) · Zbl 1364.35445 |
| [26] | Blank, L.; Garcke, H.; Sarbu, L.; Styles, V., Primal-dual active set methods for Allen-Cahnvariational inequalities with nonlocal constraints, Numer Math Partial Differential Eq, 18, 999-1030 (2013) · Zbl 1272.65060 |
| [27] | Okumura, M., A stable and structure-preserving scheme for a non-local Allen-Cahn equation, Jpn J Ind Appl Math, 35, 1245-1281 (2018) · Zbl 1403.65043 |
| [28] | Brassel, M.; Bretin, E., A modified phase-field approximation for mean curvature flow with conservation of the volume, Math Methods Appl Sci, 34, 1157-1180 (2011) · Zbl 1235.49082 |
| [29] | Alfaro, M.; Alifrangis, P., Convergence of a mass conserving Allen-Cahn equation whose lagrange multiplier is nonlocal and local, Interfaces Free Bound, 16, 243-268 (2014) · Zbl 1304.35729 |
| [30] | Golovaty, D., The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Quart Appl Math, 55, 243-298 (1997) · Zbl 0878.35059 |
| [31] | Takasao, K., Existence of weak solution for volume-preserving mean curvature flow via phase field method, Indiana U Math J, 66, 2015-2035 (2017) · Zbl 1383.53051 |
| [32] | Pazy, A., Semigroups of linear operators and applications to partial differential equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023 |
| [33] | Henry, D., Geometric theory of semilinear parabolic equations (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0456.35001 |
| [34] | Temam, R., Infinite dimensional dynamical dystems (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0871.35001 |
| [35] | Girault, V.; Raviart, P. A., Finite element methods for Navier-Stokes equations, theory and algorithms (1986), Springer-Verlag: Springer-Verlag Berlin Heidelberg New York Tokyo · Zbl 0585.65077 |
| [36] | Eyre, D. J., Unconditionally gradient stable time marching the Cahn-Hilliard equation, Mater Res Soc Symp Proc, 529, 39-46 (1998) |
| [37] | Wang, C.; Wise, S. M., An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J Numer Anal, 49, 945-969 (2011) · Zbl 1230.82005 |
| [38] | Kim, J.; Lee, S.; Choi, Y., A conservative Allen-Cahn equation with a space-time dependent lagrange multiplier, Int J Eng Sci, 84, 11-17 (2014) · Zbl 1425.65089 |
| [39] | Weng, Z.; Zhuang, Q., Numerical approximation of the conservative Allen-Cahn equation by operator splitting method, Math Meth Appl Sci, 40, 4462-4480 (2017) · Zbl 1373.65076 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
