Efficient Fully Discrete Spectral-Galerkin Scheme for the Volume-Conserved Multi-Vesicular Phase-Field Model of Lipid Vesicles with Adhesion Potential

Abstract

In this work, we aim to develop an effective fully discrete Spectral-Galerkin numerical scheme for the multi-vesicular phase-field model of lipid vesicles with adhesion potential. The essence of the scheme is to introduce several additional auxiliary variables and design some corresponding auxiliary ODEs to reformulate the system into an equivalent form so that the explicit discretization for the nonlinear terms can also achieve unconditional energy stability. Moreover, the scheme has a full decoupling structure and can avoid calculating variable-coefficient systems. The advantage of this scheme is its high efficiency and ease of implementation, that is, only by solving two independent linear biharmonic equations with constant coefficients for each phase-field variable, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. We strictly prove that the fully discrete energy stability that the scheme holds and give a detailed step-by-step implementation process. Further, numerical experiments are carried out in 2D and 3D to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

Keywords

Multi-phase-field / Lipid vesicles / Decoupled / Second-order / Volume-conserved / Unconditional energy stability

References

[1.] Aland S, Egerer S, Lowengrub J, Voigt A. Diffuse interface models of locally inextensible vesicles in a viscous fluid. J. Comput. Phys.. 2014, 277 32-47

[2.] Campelo F, Hernandez-Machado A. Shape instabilities in vesicles: a phase-field model. Eur. Phys. J. Spec. Top.. 2007, 143 101-108

[3.] Chen C, Yang X. Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn–Hilliard model. Comput. Methods Appl. Mech. Eng.. 2019, 351 35-59

[4.] Chen Q, Shen J. Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model. SIAM J. Sci. Comput.. 2018, 40 A3982-A4006

[5.] Du Q, Li M, Liu C. Analysis of a phase field Navier–Stokes vesicle-fluid interaction model. Discrete Cont. Dyn. Syst. B. 2007, 8 3 539-556

[6.] Du Q, Liu C, Ryham R, Wang X. A phase field formulation of the Willmore problem. Nonlinearity. 2005, 18 1249-1267

[7.] Du Q, Liu C, Wang X. A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys.. 2004, 198 450-468

[8.] Du Q, Zhang J. Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations. SIAM J. Sci. Comput.. 2008, 30 3 1634-1657

[9.] Du Q, Zhu L. Analysis of a mixed finite element method for a phase field bending elasticity model of vesicle membrane deformation. J. Comput. Math.. 2006, 24 265-280

[10.] Funkhouser C, Solis F, Thorton K. Coupled composition–deformation phase-field method for multicomponent lipid membranes. Phys. Rev. E. 2007, 76 011912

[11.] Gu R, Wang X, Gunzburger M. Simulating vesicle-substrate adhesion using two phase field functions. J. Comput. Phys.. 2014, 275 626-641

[12.] Gu R, Wang X, Gunzburger M. A two phase field model for tracking vesicle–vesicle adhesion. J. Math. Biol.. 2016, 73 1293-1319

[13.] Guillen-Gonzalez F, Tierra G. Unconditionally energy stable numerical schemes for phase-field vesicle membrane model. J. Comput. Phys.. 2018, 354 67-85

[14.] Li X, Li T, Tu R, Pan K, Chen C, Yang X. Efficient energy stable scheme for volume-conserved phase-field elastic bending energy model of lipid vesicles. J. Comput. Appl. Math.. 2021, 385 113177

[15.] Lowengrub J, Ratz A, Voigt A. Phase-field modeling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E. 2009, 79 031926

[16.] Marth W, Aland S, Voigt A. Margination of white blood cells: a computational approach by a hydrodynamic phase field model. J. Fluid Mech.. 2016, 790 389-406

[17.] Rubinstein J, Sternberg P. Nonlocal reaction–diffusion equations and nucleation. IMA J. Appl. Math.. 1992, 48 249-264

[18.] Shen J. Efficient spectral-Galerkin method I. Direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput.. 1994, 15 1489-1505

[19.] Shen J, Yang X. The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems. Contemp. Math.. 2020, 754 217-245

[20.] Siegel D, Kozlov M. The Gaussian curvature elastic modulus of N-monomethylated dioleoylphosphatidylethanolamine: Relevance to membrane fusion and lipid phase behavior. Biophys. J.. 2004, 87 366-374

[21.] Wang X, Du Q. Modelling and simulations of multi-component lipid membranes and open membranes via diffusive interface approaches. J. Math. Biol.. 2008, 56 347-371

[22.] Wang X, Ju L, Du Q. Efficient and stable exponential time differencing Runge–Kutta methods for phase field elastic bending energy models. J. Comput. Phys.. 2016, 316 21-38

[23.] Yang X. Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys.. 2016, 327 294-316

[24.] 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.. 2021, 376 13589

[25.] Yang X. A novel fully-decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier–Stokes equations coupled with mass-conserved Allen–Cahn phase-field model of two-phase incompressible flow. Int. J. Numer. Methods Eng.. 2021, 122 1283-1306

[26.] Yang X. A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen–Cahn type flow-coupled binary surfactant model. Comput. Methods Appl. Mech. Eng.. 2021, 373 113502

[27.] Yang X. A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model. J. Comput. Phys.. 2021, 432 110015

[28.] Yang X. A novel second-order time marching scheme for the Navier–Stokes/Darcy coupled with mass-conserved Allen–Cahn phase-field models of two-phase incompressible flow. Comput. Methods Appl. Mech. Eng.. 2021, 377 113597

[29.] Yang X. Fully-discrete spectral-Galerkin scheme with decoupled structure and second-order time accuracy for the anisotropic phase-field dendritic crystal growth model. Int. J. Heat Mass Transf.. 2021, 180 121750

[30.] Yang X. Numerical approximations of the Navier–Stokes equation coupled with volume-conserved multi-phase-field vesicles system: fully-decoupled, linear, unconditionally energy stable and second-order time-accurate numerical scheme. Comput. Methods Appl. Mech. Eng.. 2021, 375 113600

[31.] Yang X. On a novel full decoupling, linear, second-order accurate, and unconditionally energy stable numerical scheme for the anisotropic phase-field dendritic crystal growth model. Int. J. Numer. Methods Eng.. 2021, 122 4129-4153

[32.] Yang X, Ju L. Efficient linear schemes with unconditionally energy stability for the phase field elastic bending energy model. Comput. Methods Appl. Mech. Eng.. 2017, 315 691-712

[33.] Zhang G-D, He X, Yang X. Decoupled, linear, and unconditionally energy stable fully-discrete finite element numerical scheme for a two-phase ferrohydrodynamics model. SIAM J. Sci. Comput.. 2021, 43 B167-B193

[34.] Zhang J, Das S, Du Q. A phase field model for vesicle-substrate adhesion. J. Comput. Phys.. 2009, 228 7837-7849

[35.] Zhang J, Yang X. Decoupled, non-iterative, and unconditionally energy stable large time stepping method for the three-phase Cahn–Hilliard phase-field model. J. Comput. Phys.. 2020, 404 109115

[36.] Zhang J, Yang X. Unconditionally energy stable large time stepping method for the L2-gradient flow based ternary phase-field model with precise nonlocal volume conservation. Comput. Methods Appl. Mech. Eng.. 2020, 361 112743

Funding

National Science Foundation(2012490)