Three-Dimensional Lattice Boltzmann Model for the Complex Ginzburg–Landau Equation

In this paper, a lattice Boltzmann model for the three-dimensional complex Ginzburg–Landau equation is proposed. The multi-scale technique and the Chapman–Enskog expansion are used to describe higher-order moments of the complex equilibrium distribution function and a series of complex partial differential equations. The modified partial differential equation of the three-dimensional complex Ginzburg–Landau equation with the third order truncation error is obtained. Based on the complex lattice Boltzmann model, some motions of the stable scroll, such as the scroll wave with a straight filament, scroll ring, and helical scroll are simulated. The comparisons between results of the lattice Boltzmann model with those obtained by the alternative direction implicit scheme are given. The numerical results show that this model can be used to simulate the three-dimensional complex Ginzburg–Landau equation.

[1]  Bo Yan,et al.  Lattice Boltzmann Model Based on the Rebuilding-Divergency Method for the Laplace Equation and the Poisson Equation , 2011, J. Sci. Comput..

[2]  Igor S. Aranson,et al.  HELICOIDAL INSTABILITY OF A SCROLL VORTEX IN THREE-DIMENSIONAL REACTION-DIFFUSION SYSTEMS , 1998 .

[3]  S Alonso,et al.  Suppression of scroll wave turbulence by noise. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  A. R. Bishop,et al.  DYNAMICS OF VORTEX LINES IN THE THREE-DIMENSIONAL COMPLEX GINZBURG-LANDAU EQUATION : INSTABILITY, STRETCHING, ENTANGLEMENT, AND HELICES , 1998 .

[5]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results , 1993, Journal of Fluid Mechanics.

[6]  Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue. , 2007, Physical review letters.

[7]  O. Berenfeld,et al.  Shaping of a scroll wave filament by cardiac fibers. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  J. Jiménez,et al.  Boltzmann Approach to Lattice Gas Simulations , 1989 .

[9]  Succi,et al.  Extended self-similarity in the numerical simulation of three-dimensional homogeneous flows. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Shan,et al.  Lattice Boltzmann model for simulating flows with multiple phases and components. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Succi.  Numerical solution of the Schrödinger equation using discrete kinetic theory. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  D. Frenkel,et al.  Discrete solution of the electrokinetic equations. , 2004, The Journal of chemical physics.

[13]  Succi,et al.  Diffusion and hydrodynamic dispersion with the lattice Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[14]  Irina Ginzburg,et al.  Variably saturated flow described with the anisotropic Lattice Boltzmann methods , 2006 .

[15]  T. Kofané,et al.  A collective variable approach for optical solitons in the cubic–quintic complex Ginzburg–Landau equation with third-order dispersion , 2008 .

[16]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[17]  A S Mikhailov,et al.  Expanding scroll rings and negative tension turbulence in a model of excitable media. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Zhenhua Chai,et al.  A novel lattice Boltzmann model for the Poisson equation , 2008 .

[19]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

[20]  Jeffrey Yepez,et al.  QUANTUM LATTICE-GAS MODEL FOR THE DIFFUSION EQUATION , 2001 .

[21]  Peter M. A. Sloot,et al.  Lattice dependence of reaction-diffusion in lattice Boltzmann modeling , 2000 .

[22]  K. Porsezian,et al.  Modulational instability in linearly coupled complex cubic–quintic Ginzburg–Landau equations , 2009 .

[23]  J. Zhou.  Lattice Boltzmann Methods for Shallow Water Flows , 2003 .

[24]  H. Henry,et al.  Linear stability of scroll waves. , 2000, Physical review letters.

[25]  闫广武.  A lattice Boltzmann equation for waves , 2000 .

[26]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[27]  H. Verschelde,et al.  A geometric theory for scroll wave filaments in anisotropic excitable media , 2009 .

[28]  Matthaeus,et al.  Lattice Boltzmann model for simulation of magnetohydrodynamics. , 1991, Physical review letters.

[29]  Guangwu Yan,et al.  Lattice Boltzmann model for the complex Ginzburg-Landau equation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[31]  K. M. Bryden,et al.  Parallel performance and accuracy of lattice Boltzmann and traditional finite difference methods for solving the unsteady two-dimensional Burger's equation , 2006 .

[32]  C. Shu,et al.  Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Hogeweg,et al.  Scroll breakup in a three-dimensional excitable medium. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[34]  A. Winfree.  Spiral Waves of Chemical Activity , 1972, Science.

[35]  S Succi,et al.  Quantum lattice Boltzmann simulation of expanding Bose-Einstein condensates in random potentials. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Juergen Kurths,et al.  Turbulence control by developing a spiral wave with a periodic signal injection in the complex Ginzburg-Landau equation. , 2002 .

[37]  G. Vahala,et al.  Quantum Lattice Representations for Vector Solitons in External Potentials , 2006 .

[38]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : notes added in 1951 , 1951 .

[39]  J. Yepez,et al.  Quantum Lattice-Gas Model for the Burgers Equation , 2002 .

[40]  Sauro Succi,et al.  Lattice Boltzmann equation for quantum mechanics , 1993, comp-gas/9304002.

[41]  Guangwu Yan,et al.  A lattice Boltzmann model for the Korteweg-de Vries equation with two conservation laws , 2009, Comput. Phys. Commun..

[42]  Chen Yaosong,et al.  Simple lattice Boltzmann model for simulating flows with shock wave , 1999 .

[43]  M. Bär,et al.  Scroll wave instabilities in an excitable chemical medium. , 2008, Physical review letters.

[44]  Theory of the lattice boltzmann method: lattice boltzmann models for nonideal gases , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[45]  O. Steinbock,et al.  Nucleation and collapse of scroll rings in excitable media. , 2006, Physical review letters.

[46]  S. Wolfram.  Cellular automaton fluids 1: Basic theory , 1986 .

[47]  S. Hyodo,et al.  Evaluation of a lattice Boltzmann method in a complex nanoflow. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[48]  K. T. ten Tusscher,et al.  Eikonal formulation of the minimal principle for scroll wave filaments. , 2004, Physical review letters.

[49]  Ping Dong,et al.  Lattice Boltzmann schemes for the nonlinear Schrödinger equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  E. Ott,et al.  Stability of spiral wave vortex filaments with phase twists , 1998 .

[51]  John Abraham,et al.  Three-dimensional multi-relaxation time (MRT) lattice-Boltzmann models for multiphase flow , 2007, J. Comput. Phys..

[52]  Linda Vahala,et al.  Twisting of filamentary vortex solitons demarcated by fast Poincaré recursion , 2009, Defense + Commercial Sensing.

[53]  Sauro Succi,et al.  Mesoscopic lattice boltzmann modeling of flowing soft systems. , 2008, Physical review letters.

[54]  G. Vahala,et al.  Inelastic vector soliton collisions: a lattice–based quantum representation , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[55]  Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  A. Panfilov,et al.  Negative filament tension at high excitability in a model of cardiac tissue. , 2008, Physical review letters.

[57]  Sauro Succi.  Lattice Quantum Mechanics: An Application to Bose–Einstein Condensation , 1998 .

[58]  A. Panfilov,et al.  Scroll waves meandering in a model of an excitable medium. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  Jeffrey Yepez,et al.  Relativistic Path Integral as a Lattice-based Quantum Algorithm , 2005, Quantum Inf. Process..

[60]  Bastien Chopard,et al.  Lattice Boltzmann Computations and Applications to Physics , 1999, Theor. Comput. Sci..

[61]  Markus Bär,et al.  Comment on "Antispiral waves in reaction-diffusion systems". , 2004, Physical review letters.

[62]  A. Winfree.  When time breaks down , 1987 .

[63]  David J Christini,et al.  Antispiral waves in reaction-diffusion systems. , 2003, Physical review letters.

[64]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation , 1993, Journal of Fluid Mechanics.

[65]  Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  Diffusion-Induced Vortex Filament Instability in 3-Dimensional Excitable Media , 1999, patt-sol/9909004.

[67]  S. Succi,et al.  Three-Dimensional Flows in Complex Geometries with the Lattice Boltzmann Method , 1989 .

[68]  Guangwu Yan,et al.  A higher-order moment method of the lattice Boltzmann model for the Korteweg-de Vries equation , 2009, Math. Comput. Simul..

[69]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[70]  G. Vahala,et al.  Quantum lattice gas representation of some classical solitons , 2003 .

[71]  Scroll waves in the presence of slowly varying anisotropy with application to the heart. , 2001, Physical review letters.

[72]  Xijun Yu,et al.  Two-dimensional lattice Boltzmann model for compressible flows with high Mach number , 2008, 0801.4169.

[73]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[74]  Shiyi Chen,et al.  Lattice Boltzmann computations for reaction‐diffusion equations , 1993 .

[75]  Sauro Succi,et al.  Lattice Boltzmann-Poisson method for electrorheological nanoflows in ion channels , 2005, Comput. Phys. Commun..

[76]  The dynamics of scroll wave filaments in the complex Ginzburg-Landau equation , 1998 .

[77]  Guangwu Yan,et al.  Lattice Bhatnagar—Gross—Krook model for the Lorenz attractor , 2001 .

[78]  Yabi Wu,et al.  Control of spiral turbulence by periodic forcing in a reaction-diffusion system with gradients. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[79]  R. Benzi,et al.  Lattice Gas Dynamics with Enhanced Collisions , 1989 .

[80]  Rahul Kekre,et al.  Comparison of lattice-Boltzmann and brownian-dynamics simulations of polymer migration in confined flows. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[81]  Linda Vahala,et al.  Lattice Boltzmann and quantum lattice gas representations of one-dimensional magnetohydrodynamic turbulence , 2003 .

[82]  Hong Zhang,et al.  Spatiotemporal chaos control with a target wave in the complex Ginzburg-Landau equation system. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[83]  G. Vahala,et al.  MHD Turbulence Studies Using Lattice Boltzmann Algorithms-Physical Simulations Using 9,000 Cores on the Air Force Research Laboratory HAWK Supercomputer , 2008, 2008 DoD HPCMP Users Group Conference.

[84]  Mark-Anthony Bray,et al.  Interaction dynamics of a pair of vortex filament rings. , 2003, Physical review letters.

[85]  Baochang Shi,et al.  Lattice Boltzmann Simulation of Some Nonlinear Complex Equations , 2007, International Conference on Computational Science.

[86]  M. Cates,et al.  Run-and-tumble particles with hydrodynamics: sedimentation, trapping, and upstream swimming. , 2010, Physical review letters.

[87]  Guangwu Yan,et al.  Lattice Boltzmann model for wave propagation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[88]  Moran Wang,et al.  Roughness and cavitations effects on electro-osmotic flows in rough microchannels using the lattice Poisson-Boltzmann methods , 2007, J. Comput. Phys..

[89]  S Succi,et al.  Ground-state computation of Bose-Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[90]  Linda Vahala,et al.  Lattice Quantum Algorithm for the Schrödinger Wave Equation in 2+1 Dimensions with a Demonstration by Modeling Soliton Instabilities , 2005, Quantum Inf. Process..

[91]  Arkady M. Pertsov,et al.  Dynamics of scroll rings in a parameter gradient , 1999 .

[92]  Y. Kwon,et al.  Application of lattice Boltzmann method, finite element method, and cellular automata and their coupling to wave propagation problems , 2008 .

[93]  Luo Li-Shi,et al.  Theory of the lattice Boltzmann method: Lattice Boltzmann models for non-ideal gases , 2001 .

[94]  Y. Pomeau,et al.  Lattice-gas automata for the Navier-Stokes equation. , 1986, Physical review letters.

[95]  S Succi,et al.  Fast lattice Boltzmann solver for relativistic hydrodynamics. , 2010, Physical review letters.

[96]  Vincent Hakim,et al.  Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[97]  O. Filippova,et al.  Lattice-Boltzmann simulation of gas-particle flow in filters , 1997 .

[98]  Gang Hu,et al.  Suppression of Winfree turbulence under weak spatiotemporal perturbation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[99]  E. Ott,et al.  Motion of Scroll Wave Filaments in the Complex Ginzburg-Landau Equation , 1997 .

[100]  V. Biktashev,et al.  Control of scroll-wave turbulence using resonant perturbations. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[101]  Miki Hirabayashi,et al.  The lattice BGK model for the Poisson equation , 2001 .

[102]  Chenghai Sun,et al.  Lattice-Boltzmann models for high speed flows , 1998 .

[103]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[104]  H. Henry.  Spiral wave drift in an electric field and scroll wave instabilities. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[105]  Jeffrey Yepez,et al.  Simulation of the diffusion equation on a type-II quantum computer , 2002 .

[106]  Jeffrey Yepez,et al.  An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation , 2002 .

[107]  G. Vahala,et al.  Vortex-antivortex pair in a Bose-Einstein condensate , 2009 .

[108]  D. Barkley.  A model for fast computer simulation of waves in excitable media , 1991 .

[109]  Guangwu Yan,et al.  A lattice Boltzmann model for the nonlinear Schrödinger equation , 2007 .

[110]  Frisch,et al.  Lattice gas automata for the Navier-Stokes equations. a new approach to hydrodynamics and turbulence , 1989 .

[111]  Kenneth Showalter,et al.  Chemical waves and patterns , 1995 .