A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation

Abstract We present a space–time fractional Allen–Cahn phase-field model that describes the transport of the fluid mixture of two immiscible fluid phases. The space and time fractional order parameters control the sharpness and the decay behavior of the interface via a seamless transition of the parameters. Although they are shown to provide more accurate description of anomalous diffusion processes and sharper interfaces than traditional integer-order phase-field models do, fractional models yield numerical methods with dense stiffness matrices. Consequently, the resulting numerical schemes have significantly increased computational work and memory requirement. We develop a lossless fast numerical method for the accurate and efficient numerical simulation of the space–time fractional phase-field model. Numerical experiments shows the utility of the fractional phase-field model and the corresponding fast numerical method.

[1]  D. L. Young,et al.  Simulations of multidimensional interfacial flows by an improved volume-of-fluid method , 2013 .

[2]  Aranson,et al.  Continuum field description of crack propagation , 2000, Physical review letters.

[3]  Giulio Schimperna,et al.  Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations , 2015, 1502.06383.

[4]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

[5]  Masao Doi,et al.  Soft Matter Physics , 2013 .

[6]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[7]  Cheng Wang,et al.  A Linear Energy Stable Scheme for a Thin Film Model Without Slope Selection , 2012, J. Sci. Comput..

[8]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

[9]  Oleg D. Lavrentovich,et al.  Soft Matter Physics: An Introduction , 2002 .

[10]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .

[11]  T. Lubensky,et al.  Principles of condensed matter physics , 1995 .

[12]  G. Karniadakis,et al.  A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations , 2016 .

[13]  Vittorio Cristini,et al.  Drop breakup in three-dimensional viscous flows , 1998 .

[14]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

[15]  Qing Nie,et al.  Nonlinear three-dimensional simulation of solid tumor growth , 2007 .

[16]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[17]  M. Meerschaert,et al.  Stochastic Models for Fractional Calculus , 2011 .

[18]  A. Karma,et al.  Phase-field model of mode III dynamic fracture. , 2001, Physical review letters.

[19]  Howard Brenner,et al.  Interfacial transport processes and rheology , 1991 .

[20]  V. Cristini,et al.  Nonlinear simulation of tumor growth , 2003, Journal of mathematical biology.

[21]  Jie Shen,et al.  A Phase-Field Model and Its Numerical Approximation for Two-Phase Incompressible Flows with Different Densities and Viscosities , 2010, SIAM J. Sci. Comput..

[22]  Akira Onuki,et al.  Phase-field model of solid-liquid phase transition with density difference and latent heat in velocity and elastic fields. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Gautam Biswas,et al.  A coupled level-set and volume-of-fluid method for the buoyant rise of gas bubbles in liquids , 2013 .

[24]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[25]  David Jacqmin,et al.  Contact-line dynamics of a diffuse fluid interface , 2000, Journal of Fluid Mechanics.

[26]  Jia Zhao,et al.  A 3D Hydrodynamic Model for Cytokinesis of Eukaryotic Cells , 2014 .

[27]  Qiang Du,et al.  Diffuse-interface description of strain-dominated morphology of critical nuclei in phase transformations , 2008 .

[28]  I. Turner,et al.  Numerical Approximation of a Fractional-In-Space Diffusion Equation, I , 2005 .

[29]  Qiang Zhang,et al.  Three-Dimensional Front Tracking , 1998, SIAM J. Sci. Comput..

[30]  Jie Shen,et al.  Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy , 2012, SIAM J. Numer. Anal..

[31]  Xiaofeng Yang,et al.  A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals , 2017 .

[32]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[33]  A. Karma,et al.  Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Howard H. Hu,et al.  Direct numerical simulations of fluid-solid systems using the arbitrary Langrangian-Eulerian technique , 2001 .

[35]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[36]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[37]  I. Podlubny.  Fractional differential equations , 1998 .

[38]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[39]  Jia Zhao,et al.  A 3D Multi-Phase Hydrodynamic Model for Cytokinesis of Eukaryotic Cells , 2016 .

[40]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[41]  Cheng Wang,et al.  Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation , 2009, J. Comput. Phys..

[42]  Mark Sussman,et al.  A sharp interface method for incompressible two-phase flows , 2007, J. Comput. Phys..

[43]  Hong Wang,et al.  A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations , 2013, J. Comput. Phys..

[44]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[45]  George Keith Batchelor,et al.  An Introduction to Fluid Dynamics. , 1969 .

[46]  Cv Clemens Verhoosel,et al.  A phase-field description of dynamic brittle fracture , 2012 .

[47]  Britta Nestler,et al.  Crystal growth of pure substances: Phase-field simulations in comparison with analytical and experimental results , 2005 .

[48]  H. Frieboes,et al.  Nonlinear modelling of cancer: bridging the gap between cells and tumours , 2010, Nonlinearity.