Error estimate of full-discrete numerical scheme for the nonlocal Allen-Cahn model. (English) Zbl 07892713
Summary: In this work, we study the error estimates of the fully discrete Fourier pseudo-spectral numerical scheme for solving the nonlocal volume-conserved Allen-Cahn (AC) equation. The time marching method of the numerical scheme is based on the well-known Invariant Energy Quadratization (IEQ) method. We demonstrate that the proposed fully discrete numerical method is uniquely solvable, unconditionally energy stable, and obtain the optimal error estimate of the scheme for both space and time. Additionally, we conduct several numerical tests to verify the theoretical results.
MSC:
| 35K45 | Initial value problems for second-order parabolic systems |
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 65G20 | Algorithms with automatic result verification |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlocal Allen-Cahn model; uniquely solvable; unconditionally energy stable; error estimate; numerical testsReferences:
| [1] | J. RUBINSTEIN, P. STERNBERG. Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math., 1992, 48(3): 249-264. · Zbl 0763.35051 |
| [2] | J. LOWENGRUB, L. TRUSKINOVSKY. Quasi-incompressible Cahn-Hilliard fluids and topological transi-tions. · Zbl 0927.76007 |
| [3] | R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 1998, 454(1978): 2617-2654. · Zbl 0927.76007 |
| [4] | D. M. ANDERSON, G. B. MCFADDEN, A. A. WHEELER. Diffuse-Interface Methods in Fluid Mechanics. Annual Reviews, Palo Alto, CA, 1998. · Zbl 1398.76051 |
| [5] | Chun LIU, Jie SHEN. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D, 2003, 179(3-4): 211-228. · Zbl 1092.76069 |
| [6] | Yibao LI, D. JEONG, J. I. CHOI, et al. Fast local image inpainting based on the Allen-Cahn model. Digital Signal Proc., 2015, 37: 65-74. |
| [7] | Hongwei LI, Lili JU, Chenfeng ZHANG, et al. Unconditionally energy stable linear schemes for the diffuse interface model with Peng-Robinson equation of state. J. Sci. Comput., 2018, 75(2): 993-1015. · Zbl 1398.65305 |
| [8] | Jia ZHAO, Xiaofeng YANG, Yuezheng GONG, et al. A general strategy for numerical approximations of non-equilibrium modelspart I: thermodynamical systems. Int. J. Numer. Anal. Model., 2018, 15(6): 884-918. · Zbl 1412.65099 |
| [9] | S. M. ALLEN, J. W. CAHN. Ground state structures in ordered binary alloys with second neighbor interac-tions. Acta Metallurgica, 1972, 20(3): 423-433. |
| [10] | Chuanjun CHEN, Xiaofeng YANG. Highly efficient and unconditionally energy stable semi-discrete time-marching numerical scheme for the two-phase incompressible flow phase-field system with variable-density and viscosity. Sci. China Math., 2022, 65(12): 2631-2656. · Zbl 1504.65190 |
| [11] | J. W. CAHN, J. E. HILLIARD. Free energy of a non-uniform system. I. interfacial free energy. J. Chemical Phys., 1958, 28(2): 258-267. · Zbl 1431.35066 |
| [12] | Chuanjun CHEN, Xiaofeng YANG. A second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-nematic model for two-phase complex fluids confined in the Hele-Shaw cell. J. Comput. Phys., 2022, 456: Paper No. 111026, 24 pp. · Zbl 07518104 |
| [13] | Chuanjun CHEN, Xiaofeng YANG. Fully-discrete finite element numerical scheme with decoupling structure and energy stability for the Cahn-Hilliard phase-field model of two-phase incompressible flow system with variable density and viscosity. ESAIM Math. Model. Numer. Anal., 2021, 55(5): 2323-2347. · Zbl 1491.65091 |
| [14] | Xinfu CHEN, D. HILHORST, E. LOGAK. Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound., 2010, 12(4): 527-549. · Zbl 1219.35018 |
| [15] | M. BRASSEL, E. BRETIN. A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Methods Appl. Sci., 2011, 34(10): 1157-1180. · Zbl 1235.49082 |
| [16] | M. ALFARO, P. ALIFRANGIS. Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local. Interfaces Free Bound., 2014, 16(2): 243-268. · Zbl 1304.35729 |
| [17] | Jun ZHANG, Xiaofeng YANG. Numerical approximations for a new L 2 -gradient flow based phase field crystal model with precise nonlocal mass conservation. Comput. Phys. Commun., 2019, 243: 51-67. · Zbl 07674815 |
| [18] | Jun ZHANG, Xiaofeng YANG. Unconditionally energy stable large time stepping method for the L 2 -gradient flow based ternary phase-field model with precise nonlocal volume conservation. Comput. Methods Appl. Mech. Engrg., 2020, 361: 112743, 23 pp. · Zbl 1442.65186 |
| [19] | Ziqiang WANG, Chuanjun CHEN, Yanjun LI, et al. Decoupled finite element scheme of the variable-density and viscosity phase-field model of a two-phase incompressible fluid flow system using the volume-conserved Allen-Cahn dynamics. J. Comput. Appl. Math., 2023, 420: Paper No. 114773, 16 pp. · Zbl 1498.65165 |
| [20] | Qiongwei YE, Zhigang OUYANG, Chuanjun CHEN, et al. Efficient decoupled second-order numerical scheme for the flow-coupled Cahn-Hilliard phase-field model of two-phase flows. J. Comput. Appl. Math., 2022, 405: Paper No. 113875, 16 pp. · Zbl 1524.65623 |
| [21] | Guan ZHEN, J. S. LOWENGRUB, Wang CHENG, et al. Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations. J. Comput. Phys., 2014, 277: 48-71. · Zbl 1349.65298 |
| [22] | Guan ZHEN, J. S. LOWENGRUB, Wang CHENG. Convergence analysis for second-order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations. Math. Methods Appl. Sci., 2017, 40(18): 6836-6863. · Zbl 1387.65087 |
| [23] | Shuying ZHAI, Zhifeng WENG, Xinlong FENG. Investigations on several numerical methods for the non-local Allen-Cahn equation. International J. Heat Mass Transfer, 2015, 87: 111-118. |
| [24] | Shouwen SUN, Xiaobo JING, Qi WANG. Error estimates of energy stable numerical schemes for Allen-Cahn equations with nonlocal constraints. J. Sci. Comput., 2019, 79(1): 593-623. · Zbl 1502.65075 |
| [25] | Xiaofeng YANG, Jia ZHAO, Qie WANG, et al. Numerical approximations for a three-component Cahn-Hilliard phase-field model based on the invariant energy quadratization method. Math. Models Methods Appl. Sci., 2017, 27(11): 1993-2030. · Zbl 1393.80003 |
| [26] | Xiaofeng YANG, Guodong ZHANG. Numerical approximations of the Cahn-Hilliard and Allen-Cahn equa-tions with general nonlinear potential using the invariant energy quadratization approach. arXiv:1712.02760, 2017. |
| [27] | Xiaofeng YANG, Lili JU. Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model. Comput. Methods Appl. Mech. Engrg., 2017, 315: 691-712. · Zbl 1439.74165 |
| [28] | Xiaofeng YANG, Jia ZHAO, Qi WANG. Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys., 2017, 333: 104-127. · Zbl 1375.82121 |
| [29] | Haijun YU, Xiaofeng YANG. Numerical approximations for a phase-field moving contact line model with variable densities and viscosities. J. Comput. Phys., 2017, 334: 665-686. · Zbl 1375.76201 |
| [30] | Jia ZHAO, Xiaofeng YANG, Yuezheng GONG, et al. A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals. Comput. Methods Appl. Mech. Engrg., 2017, 318: 803-825. · Zbl 1439.76124 |
| [31] | C. CANUTO, A. QUARTERONI. Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp., 1982, 38(157): 67-86. · Zbl 0567.41008 |
| [32] | Jun ZHANG, Jia ZHAO, Yuezheng GONG. Error analysis of full-discrete invariant energy quadratization schemes for the Cahn-Hilliard type equation. J. Comput. Appl. Math., 2020, 372: 112719, 15 pp. · Zbl 1440.65257 |
| [33] | Qi HONG, Yuezheng GONG, Jia ZHAO, et al. Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint. Appl. Numer. Math., 2021, 170: 321-339. · Zbl 1501.65080 |
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.
