Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations

We devise second-order accurate, unconditionally uniquely solvable and unconditionally energy stable schemes for the nonlocal Cahn-Hilliard (nCH) and nonlocal Allen-Cahn (nAC) equations for a large class of interaction kernels. We present numerical evidence that both schemes are convergent. We solve the nonlinear equations resulting from discretization using an efficient nonlinear multigrid method and demonstrate the performance of our algorithms by simulating nucleation and crystal growth for several different choices of interaction kernels.

[1]  Luciano Reatto,et al.  s of t ] 1 3 Se p 20 07 Microphase morphology in 2 D fluids under lateral confinement , 2008 .

[2]  Luciano Reatto,et al.  A bidimensional fluid system with competing interactions: spontaneous and induced pattern formation , 2004 .

[3]  Robert C. Rogers.  Some remarks on nonlocal interactions and hysteresis in phase transitions , 1994 .

[4]  H. Frieboes,et al.  Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method. , 2008, Journal of theoretical biology.

[5]  Klaus Gärtner,et al.  On a nonlocal model of image segmentation , 2005 .

[6]  Kevin J Painter,et al.  Adding Adhesion to a Chemical Signaling Model for Somite Formation , 2009, Bulletin of mathematical biology.

[7]  Pedro Tarazona,et al.  Dynamic density functional theory of fluids , 1999 .

[8]  A. Archer,et al.  Dynamical density functional theory and its application to spinodal decomposition. , 2004, The Journal of chemical physics.

[9]  Martin Grant,et al.  Modeling elasticity in crystal growth. , 2001, Physical review letters.

[10]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[11]  Gunduz Caginalp,et al.  Interface Conditions for a Phase Field Model with Anisotropic and Non-Local Interactions , 2011 .

[12]  N. Provatas,et al.  Free energy functionals for efficient phase field crystal modeling of structural phase transformations. , 2010, Physical review letters.

[13]  Peter W. Bates,et al.  On a nonlocal phase-field system , 2006 .

[14]  Zhen Guan.  Numerical Analysis of First and Second Order Unconditional Energy Stable Schemes for Nonlocal Cahn-Hilliard and Allen-Cahn Equations , 2012 .

[15]  Cheng Wang,et al.  A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation , 2014, Numerische Mathematik.

[16]  K. Howell.  Principles of Fourier Analysis , 2001 .

[17]  Giambattista Giacomin,et al.  Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits , 1997, comp-gas/9705001.

[18]  Paul C. Fife,et al.  Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions , 2003 .

[19]  Nasser Mohieddin Abukhdeir,et al.  Long-time integration methods for mesoscopic models of pattern-forming systems , 2011, J. Comput. Phys..

[20]  M. Chaplain,et al.  Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. , 2008 .

[21]  Daming Li,et al.  DDFT calibration and investigation of an anisotropic phase-field crystal model. , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

[22]  Zhonghua Qiao,et al.  An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation , 2012 .

[23]  John W. Cahn,et al.  On Spinodal Decomposition , 1961 .

[24]  Thomas Wanner,et al.  A semi-implicit spectral method for stochastic nonlocal phase-field models , 2009 .

[25]  M. Grant,et al.  Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  R. Rogers,et al.  A nonlocal model for the exchange energy in ferromagnetic materials , 1991 .

[27]  Herbert Gajewski,et al.  On a nonlocal phase separation model , 2003 .

[28]  Markos A. Katsoulakis,et al.  Spectral methods for mesoscopic models of pattern formation , 2001 .

[29]  R. Evans.  The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids , 1979 .

[30]  S. M. Wise,et al.  Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations , 2010, J. Sci. Comput..

[31]  E. Knobloch,et al.  Solidification fronts in supercooled liquids: how rapid fronts can lead to disordered glassy solids. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Joel Lebowitz,et al.  Phase Segregation Dynamics in Particle Systems with Long Range Interactions II: Interface Motion , 1997, SIAM J. Appl. Math..

[33]  J. Lowengrub,et al.  Conservative multigrid methods for Cahn-Hilliard fluids , 2004 .

[34]  Andrew J. Archer,et al.  Dynamical density functional theory for interacting Brownian particles: stochastic or deterministic? , 2004, cond-mat/0405603.

[35]  Ekkehard W. Sachs,et al.  Efficient solution of a partial integro-differential equation in finance , 2008 .

[36]  Peter W. Bates,et al.  The Dirichlet boundary problem for a nonlocal Cahn–Hilliard equation , 2005 .

[37]  Harald Garcke,et al.  Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility , 1999, SIAM J. Numer. Anal..

[38]  Peter W. Bates,et al.  The Neumann boundary problem for a nonlocal Cahn–Hilliard equation , 2005 .

[39]  L. Reatto,et al.  Microphase separation in two-dimensional systems with competing interactions. , 2006, The Journal of chemical physics.

[40]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[41]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[42]  J. Heath,et al.  Spontaneous patterning of quantum dots at the air-water interface. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[43]  Cornelis W. Oosterlee,et al.  Numerical valuation of options with jumps in the underlying , 2005 .

[44]  Vittorio Cristini,et al.  Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis. , 2010, Journal of theoretical biology.

[45]  John W. Barrett,et al.  An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy , 1995 .

[46]  Gerhard Kahl,et al.  Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density-functional theory. , 2007, The Journal of chemical physics.

[47]  Xinfu Chen,et al.  ON SOME NONLOCAL EVOLUTION EQUATIONS ARISING IN MATERIALS SCIENCE , 2008 .

[48]  Zhizhang Chen.  Unconditionally stable finite-difference time-domain methods and their applications , 2004, Proceedings. ICCEA 2004. 2004 3rd International Conference on Computational Electromagnetics and Its Applications, 2004..

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

[50]  P. Bates,et al.  International Journal of C 2009 Institute for Scientific Numerical Analysis and Modeling Computing and Information Numerical Analysis for a Nonlocal Allen-cahn Equation , 2022 .

[51]  Kun Zhou,et al.  Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints , 2012, SIAM Rev..

[52]  A. Archer,et al.  Two-dimensional fluid with competing interactions exhibiting microphase separation: theory for bulk and interfacial properties. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Enza Orlandi,et al.  Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics , 1994 .

[54]  V. Gavini,et al.  Classical density functional theory and the phase-field crystal method using a rational function to describe the two-body direct correlation function. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  van der Kg Kristoffer Zee,et al.  Stabilized second‐order convex splitting schemes for Cahn–Hilliard models with application to diffuse‐interface tumor‐growth models , 2014, International journal for numerical methods in biomedical engineering.

[56]  Cheng Wang,et al.  An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation , 2009, SIAM J. Numer. Anal..

[57]  Ioannis G Kevrekidis,et al.  Dynamic density functional theory of solid tumor growth: Preliminary models. , 2012, AIP advances.

[58]  Qiang Du,et al.  APPLICATION OF A NONLOCAL VECTOR CALCULUS TO THE ANALYSIS OF LINEAR PERIDYNAMIC MATERIALS. , 2011 .

[59]  K. Painter,et al.  A continuum approach to modelling cell-cell adhesion. , 2006, Journal of theoretical biology.

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