ORIGINAL PAPERS

Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows

Expand
  • 1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, Henan, China;
    2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received date: 2022-07-20

  Revised date: 2023-01-29

  Online published: 2024-04-16

Supported by

Research of Ruihan Guo was partially supported by the NSFC Grant no. 12271492, and the Natural Science Foundation of Henan Province of China Grant no. 222300420550. Research of Yinhua Xia was partially supported by the NSFC Grant no. 12271498, and the National Key R & D Program of China Grant no. 2022YFA1005202/2022YFA1005200.

Abstract

Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows, it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations. One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them. The resulting schemes are only first-order accurate in time, and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version. In this paper, we employ the spectral deferred correction method to improve the temporal accuracy, based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea. The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency. Within the framework of the spatially discretized local discontinuous Galerkin method, the resulting numerical schemes are fully decoupled, efficient, and high-order accurate in both time and space. Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.

Cite this article

Ruihan Guo, Yinhua Xia . Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows[J]. Communications on Applied Mathematics and Computation, 2024 , 6(1) : 625 -657 . DOI: 10.1007/s42967-023-00283-9

References

[1] Caffarelli, L.A., Muler, N.E.:An L bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 133, 129-144 (1995)
[2] Chen, C., Yang, X.:Efficient numerical scheme for a dendritic solidification phase field model with melt convection. J. Comput. Phys. 388, 41-62 (2019)
[3] Cockburn, B., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II:general framework. Math. Comp. 52, 411-435 (1989)
[4] Dong, B., Shu, C.-W.:Analysis of a local discontinuous Galerkin method for fourth-order time-depenedent problems. SIAM J. Numer. Anal. 47, 3240-3268 (2009)
[5] Dutt, A., Greengard, L., Rokhlin, V.:Spectral deferred correction methods for ordinary differential equations. BIT 40, 241-266 (2000)
[6] Eyre, D.J.:Unconditionally gradient stable time marching the Cahn-Hilliard equation. MRS Online Proceedings Library 529, 39-46 (1998)
[7] Gong, Y., Zhao, J., Wang, Q.:Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models. Comput. Phys. Commun. 249, 107033 (2020)
[8] Guermond, J.L., Minev, P., Shen, J.:An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011-6045 (2006)
[9] Guermond, J.L., Salgado, A.:A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228, 2834-2846 (2009)
[10] Guo, R., Xia, Y., Xu, Y.:An efficient fully-discrete local discontinuous Galerkin method for the Cahn-Hilliard-Hele-Shaw system. J. Comput. Phys. 264, 23-40 (2014)
[11] Guo, R., Xia, Y., Xu, Y.:Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations. J. Comput. Phys. 338, 269-284 (2017)
[12] 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)
[13] Han, D., Wang, X.:A second order in time, decoupled, unconditionally stable numerical scheme for the Cahn-Hilliard-Darcy system. J. Sci. Comput. 77, 1210-1233 (2018)
[14] Ju, L., Zhang, J., Du, Q.:Fast and accurate algorithms for simulating coarsening dynamics of Cahn-Hilliard equations. Comput. Mater. Sci. 108, 272-282 (2015)
[15] Li, X., Ju, L., Meng, X.:Convergence analysis of exponential time differencing schemes for the Cahn-Hilliard equation. Commun. Comput. Phys. 26, 1510-1529 (2019)
[16] Liu, C., Shen, J., Yang, X.:Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. J. Sci. Comput. 62, 601-622 (2015)
[17] Minion, M.L.:Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1, 471-500 (2003)
[18] Minjeaud, S.:An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer. Methods Partial Differential Equations 29, 584-618 (2013)
[19] Reed, W., Hill, T.:Triangular mesh methods for the neutron transport equation, Technical report LA-UR-73-479. Los Alamos Scientific Laboratory, Los Alamos, NM (1973)
[20] Shen, J., Xu, J., Yang, J.:The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407-416 (2017)
[21] Shen, J., Xu, J., Yang, J.:A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61, 474-506 (2019)
[22] Shen, J., Yang, X.:Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28, 1169-1691 (2010)
[23] Shen, J., Yang, X.:A phase-field model for two-phase flows with large density ratio and its numerical approximation. SIAM J. Sci. Comput. 32, 1159-1179 (2010)
[24] Shen, J., Yang, X.:Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53, 279-296 (2015)
[25] Shu, C.-W.:Discontinuous Galerkin methods:general approach and stability, numerical solutions of partial differential equations. In:Bertoluzza, S., Falletta, S., Russo, G., Shu, C.-W. (eds.) Advanced Courses in Mathematics CRM Barcelona, pp. 149-201. Birkhauser, Basel (2009)
[26] Tang, T., Qiao, Z.:Efficient numerical methods for phase-field equations. Scientia Sinica Mathematica 48, 1-20 (2020)
[27] 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)
[28] Xu, C., Tang, T.:Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44, 1759-1779 (2006)
[29] Xu, Y., Shu, C.-W.:Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1-46 (2010)
[30] Yan, F., Xu, Y.:Stability analysis and error estimates of local discontinuous Galerkin method with semi-implicit spectral deferred correction time-marching for the Allen-Cahn equation. J. Comput. Appl. Math. 376, 112857 (2020)
[31] Yang, X.:Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294-316 (2016)
[32] Yang, X.:A new efficient fully-decoupled and second-order time-accurate scheme for Cahn-Hilliard phase-field model of three-phase incompressible flow. Comput. Methods Appl. Mech. Eng. 376, 113589 (2021)
[33] Yang, X.:On a novel fully-decoupled, linear and second-order accurate numerical scheme for the Cahn-Hilliard-Darcy system of two-phase Hele-Shaw flow. Comput. Phys. Commun. 263, 107868 (2021)
[34] Yang, X., He, X.:A fully-discrete decoupled finite element method for the conserved Allen-Cahn type phase-field model of three-phase fluid flow system. Comput. Methods Appl. Mech. Eng. 389, 114376 (2022)
[35] Zhao, J., Han, D.:Second-order decoupled energy-stable schemes for Cahn-Hilliard-Navier-Stokes equations. J. Comput. Phys. 443, 110536 (2021)
[36] Zhou, L., Xu, Y.:Stability analysis and error estimates of semi-implicit spectral deferred correction coupled with local discontinuous Galerkin method for linear convection-diffusion equations. J. Sci. Comput. 77, 1001-1029 (2018)
[37] Zhu, J., Chen, L.Q., Shen, J., Tikare, V.:Morphological evolution during phase separation and coarsening with strong inhomogeneous elasticity. Model. Simul. Mater. Sci. Eng. 9, 499-511 (2001)
Options
Outlines

/