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Journal of Scientific Computing

Erschienen in:

01.09.2014

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

verfasst von: Jianying Zhang, Guangwu Yan

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2014

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Abstract

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.

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Higuera, F.J., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. Europhys. Lett. 9, 345–349 (1989)

Higuera, F.J., Jimènez, J.: Boltzmann approach to lattice gas simulations. Europhys. Lett. 9, 663–668 (1989)

Qian, Y.H., d’humieres, D., Lallemand, P.: Lattice BGK model for Navier–Stokes equations. Europhys. Lett. 17(6), 479–484 (1992)MATH

Chen, H.D., Chen, S.Y., Matthaeus, M.H.: Recovery of the Navier–Stokes equations using a lattice Boltzmann gas method. Phys. Rev. A 45, 5339–5342 (1992)

Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equations: theory and applications. Phys. Rep. 222, 145–197 (1992)

Chen, S.Y., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Fluid Mech. 3, 314–322 (1998)

Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, New York (2001)MATH

Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice gas automata for the Navier–Stokes equations. Phys. Rev. Lett. 56, 1505–1508 (1986)

Wolfram, S.: Cellular automaton fluids 1: basic theory. J. Stat. Phys. 45, 471–518 (1986)MATHMathSciNet

10.

Shan, X.W., Chen, H.D.: Lattice Boltzmann model of simulating flows with multiple phases and components. Phys. Rev. E 47, 1815–1819 (1993)

11.

Luo, L.S.: Theory of the lattice Boltzmann method: lattice Boltzmann method for nonideal gases. Phys. Rev. E 62, 4982–4996 (2000)MathSciNet

12.

Premnath, K.N., Abraham, J.: Three-dimensional multi-relaxation lattice Boltzmann models for multiphase flows. J. Comput. Phys. 224, 539–559 (2007)MATHMathSciNet

13.

Ladd, A.: Numerical simulations of particle suspensions via a discretized Boltzmann equation, part 2. Numerical results. J. Fluids Mech. 271, 311–339 (1994)MathSciNet

14.

Filippova, O., Hanel, D.: Lattice Boltzmann simulation of gas-particle flow in filters. Comput. Fluids 26, 697–712 (1997)MATH

15.

Chen, S.Y., Chen, H.D., Martinez, D., et al.: Lattice Boltzmann Model for simulation of magneto-hydrodynamics. Phys. Rev. Lett. 67, 3776–3779 (1991)

16.

Vahala, L., Vahala, G., Yepez, J.: Lattice Boltzmann and quantum lattice gas representations of one-dimensional magnetohydrodynamic turbulence. Phys. Lett. A 306, 227–234 (2003)MATHMathSciNet

17.

Vahala, G., Keating, B., Soe, M., et al.: MHD turbulence studies using lattice Boltzmann algorithms. Commun. Comput. Phys. 4, 624–646 (2008)

18.

Dawson, S.P., Chen, S.Y., Doolen, G.D.: Lattice Boltzmann computations for reaction–diffusion equations. J. Chem. Phys. 98, 1514–1523 (1993)

19.

Yepez, J.: Quantum lattice-gas model for the diffusion equation. Int. J. Mod. Phys. C 12, 1285–1303 (2001)MathSciNet

20.

Berman, G.P., Ezhov, A.A., Kamenev, D.I., et al.: Simulation of the diffusion equation on a type-II quantum computer. Phys. Rev. A 66, 012310 (2002)

21.

Cali, A., Succi, S., Cancelliere, A., et al.: Diffusion and hydrodynamic dispersion with the lattice Boltzmann method. Phys. Rev. A 45, 5771–5774 (1992)

22.

Blaak, R., Sloot, P.M.: Lattice dependence of reaction–diffusion in lattice Boltzmann modeling. Comput. Phys. Commun. 129, 256–266 (2000)MATHMathSciNet

23.

Succi, S., Foti, E., Higuera, F.J.: 3-Dimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett. 10, 433–438 (1989)

24.

Sun, C.H.: Lattice-Boltzmann model for high speed flows. Phys. Rev. E 58, 7283–7287 (1998)

25.

Yan, G.W., Chen, Y.S., Hu, S.X.: Simple lattice Boltzmann model for simulating flows with shock wave. Phys. Rev. E 59, 454–459 (1999)

26.

Qu, K., Shu, Q., Chew, Y.T.: Alternative method to construct equilibrium distribution function in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number. Phys. Rev. E 75, 036706 (2007)MathSciNet

27.

Gan, Y.B., Xu, A.G., Zhang, G.C., et al.: Two-dimensional lattice Boltzmann model for compressible flows with high Mach number. Phys. A 387, 1721–1732 (2008)

28.

Mendoza, M., Boghosian, B.M., Herrmann, H.J., Succi, S.: Fast lattice Boltzmann solver for relativistic hydrodynamics. Phys. Rev. Lett. 105, 014502 (2010)

29.

Nash, R.W., Adhikari, R., Tailleur, J., Cates, M.E.: Run-and-tumble particles with hydrodynamics: sedimentation, trapping, and upstream swimming. Phys. Rev. Lett. 104, 258101 (2010)

30.

Benzi, R., Chibbaro, S., Succi, S.: Mesoscopic lattice Boltzmann modeling of flowing soft systems. Phys. Rev. Lett. 102, 026002 (2009)

31.

Li, H., Ki, H.: Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas. Phys. Rev. E 82, 016703 (2010)

32.

Suga, K., Takenaka, S., Ito, T., et al.: Evaluation of a lattice Boltzmann method in a complex nanoflow. Phys. Rev. E 82, 016701 (2010)

33.

Kekre, R., Butler, J.E., Ladd, A.: Comparison of lattice-Boltzmann and Brownian-dynamics simulations of polymer migration in confined flows. Phys. Rev. E 82, 011802 (2010)

34.

Chopard, B., Luthi, P.O.: Lattice Boltzmann computations and applications to physics. Theor. Comput. Sci. 217, 115–130 (1999)MATHMathSciNet

35.

Yan, G.W.: A lattice Boltzmann equation for waves. J. Comput. Phys. 161, 61–69 (2000)MATHMathSciNet

36.

Zhang, J.Y., Yan, G.W., Shi, X.B.: Lattice Boltzmann model for wave propagation. Phys. Rev. E 80, 026706 (2009)

37.

Kwon, Y.W., Hosoglu, S.: Application of lattice Boltzmann method, finite element method, and cellular automata and their coupling to wave propagation problems. Comput. Struct. 86, 663–670 (2008)

38.

Yepez, J.: Quantum lattice-gas model for the Burgers equation. J. Stat. Phys. 107, 203–224 (2002)MATH

39.

Yepez, J.: Open quantum system model of the one-dimensional Burgers equation with tunable shear viscosity. Phys. Rev. A 74, 042322 (2006)

40.

Velivelli, A.C., Bryden, K.M.: Parallel performance and accuracy of lattice Boltzmann and traditional finite difference methods for solving the unsteady two-dimensional Burger’s equation. Phys. A 362, 139–145 (2006)

41.

Vahala, G., Yepez, J., Vahala, L.: Quantum lattice gas representation of some classical solitons. Phys. Lett. A 310, 187–196 (2003)MATHMathSciNet

42.

Yan, G.W., Zhang, J.Y.: A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation. Math. Comput. Simul. 79, 1554–1565 (2009)MATH

43.

Zhang, J.Y., Yan, G.W.: A lattice Boltzmann model for the Korteweg–de Vries equation with two conservation laws. Comput. Phys. Commun. 180, 1054–1062 (2009)MATH

44.

Yan, G.W., Yuan, L.: Lattice Bhatnagar–Gross–Krook model for the Lorenz attractor. Phys. D 154, 43–50 (2001)MATHMathSciNet

45.

Zhou, J.G.: Lattice Boltzmann Methods for Shallow Water Flows. Springer, Berlin (2000)

46.

Ginzburg, I.: Variably saturated flow described with the anisotropic lattice Boltzmann methods. J. Comput. Fluids 25, 831–848 (2006)

47.

Melchionna, S., Succi, S.: Lattice Boltzmann–Poisson method for electrorheological nanoflows in ion channels. Comput. Phys. Commun. 169, 203–206 (2005)MATH

48.

Capuani, F., Pagonabarraga, I., Frenkel, D.: Discrete solution of the electrokinetic equations. J. Chem. Phys. 121, 973–986 (2004)

49.

Hirabayashi, M., Chen, Y., Ohashi, H.: The lattice BGK model for the Poisson equation. JSME Int. J. Ser. B 44, 45–52 (2001)

50.

Chai, Z.H., Shi, B.C.: A novel lattice Boltzmann model for the Poisson equation. Appl. Math. Model. 32, 2050–2058 (2008)MATHMathSciNet

51.

Wang, M.R., Wang, J.K., Chen, S.Y.: Roughness and cavitations effect on electro-osmotic flows in rough microchannels using the lattice Poisson–Boltzmann methods. J. Comput. Phys. 226, 836–851 (2007)MATH

52.

Wang, H.M., Yan, G.W., Yan, B.: Lattice Boltzmann model based on the rebuilding-divergency method for the Laplace equation and the Poisson equation. J. Sci. Comput. 46, 470–484 (2010)

53.

Succi, S., Benzi, R.: Lattice Boltzmann equation for quantum mechanics. Phys. D 69, 327–332 (1993)MATHMathSciNet

54.

Succi, S.: Lattice quantum mechanics: an application to Bose–Einstein condensation. Int. J. Mod. Phys. C 9, 1577–1585 (1998)

55.

Zhong, L.H., Feng, S.D., Dong, P., Gao, S.T.: Lattice Boltzmann schemes for the nonlinear Schrödinger equation. Phys. Rev. E 74, 036704 (2006)

56.

Zhang, J.Y., Yan, G.W.: A lattice Boltzmann model for the nonlinear Schrödinger equation. J. Phys. A 40, 10393–10405 (2007)MATHMathSciNet

57.

Shi, B.C.: Lattice Boltzmann simulation of some nonlinear complex equations. Lect. Notes Comput. Sci. 4487, 818–825 (2007)

58.

Yepez, J., Vahala, G., Vahala, L.: Vortex–antivortex pair in a Bose–Einstein condensate. Eur. Phys. J. Spec. Top. 171, 9–14 (2009)

59.

Yepez, J., Vahala, G., Vahala, L.: Twisting of filamentary vortex solitons demarcated by fast Poincaré recursion. Proc. SPIE 7342, 73420M (2009)

60.

Yepez, J., Vahala, G., Vahala, L.: Lattice quantum algorithm for the Schrödinger wave equation in 2 + 1 dimensions with a demonstration by modeling soliton instabilities. Quantum Inf. Process. 4, 457–469 (2005)

61.

Vahala, G., Vahala, L., Yepez, J.: Inelastic vector soliton collisions: a lattice-based quantum representation. Philos. Trans. R. Soc. A 362, 1677–1690 (2004)MATHMathSciNet

62.

Vahala, G., Vahala, L., Yepez, J.: Quantum lattice representations for vector solitons in external potentials. Phys. A 362, 215–221 (2006)

63.

Succi, S.: Numerical solution of the Schrödinger equation using discrete kinetic theory. Phys. Rev. E 53, 1969–1975 (1996)

64.

Yepez, J., Boghosian, B.: An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation. Comput. Phys. Commun. 146, 280–294 (2002)MATHMathSciNet

65.

Palpacelli, S., Succi, S., Spigler, R.: Ground-state computation of Bose–Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme. Phys. Rev. E 76, 036712 (2007)

66.

Palpacelli, S., Succi, S.: Quantum lattice Boltzmann simulation of expanding Bose–Einstein condensates in random potentials. Phys. Rev. E 77, 066708 (2008)

67.

Yepez, J.: Relativistic path integral as a lattice-based quantum algorithm. Quantum Inf. Process. 4, 471–509 (2005)MathSciNet

68.

Zhang, J.Y., Yan, G.W.: Lattice Boltzmann model for the complex Ginzburg–Landau equation. Phys. Rev. E 81, 066705 (2010)MathSciNet

69.

Aranson, I.S., Kramer, L.: The world of the complex Ginzburg–Landau equation. Rev. Mod. Phys. 74, 99–143 (2002)MATHMathSciNet

70.

Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984)MATH

71.

Winfree, A.T.: Spiral waves of chemical activity. Science 175, 634–636 (1972)

72.

Fewo, S.I., Kofane, T.C.: A collective variable approach for optical solitons in the cubic–quintic complex Ginzburg–Landau equation with third-order dispersion. Opt. Commun. 281, 2893–2906 (2008)

73.

Porsezian, K., Murali, R., Malomed, B.A., et al.: Modulational instability in linearly coupled complex cubic–quintic Ginzburg–Landau equations. Chaos Solitons Fractals 40, 1907–1913 (2009)MATHMathSciNet

74.

Jiang, M.X., Wang, X.N., Ouyang, Q., et al.: Spatiotemporal chaos control with a target wave in the complex Ginzburg–Landau equation system. Phys. Rev. E 69, 056202 (2004)

75.

Zhang, S.L., Bambi, H., Zhang, H.: Analytical approach to the drift of the tips of spiral waves in the complex Ginzburg–Landau equation. Phys. Rev. E 67, 016214 (2003)MathSciNet

76.

Gong, Y.F., Christini, D.J.: Antispiral waves in reaction–diffusion systems. Phys. Rev. Lett. 90, 088302 (2003)

77.

Brusch, L., Nicola, M.E., Bär, M.: Comment on antispiral waves in reaction–diffusion systems. Phys. Rev. Lett. 92, 89801 (2004)

78.

Kapral, R., Showalter, K.: Chemical Waves and Patterns. Kluwer, Dordrecht (1995)

79.

Winfree, A.T.: When Time Breaks Down. Princeton University Press, NJ (1987)

80.

Berenfeld, O., Wellner, M., Jalife, J., Pertsov, A.M.: Shaping of a scroll wave filament by cardiac fibers. Phys. Rev. E 63, 061901 (2001)

81.

Morgan, S.W., Biktasheva, I.V., Biktashev, V.N.: Control of scroll-wave turbulence using resonant perturbations. Phys. Rev. E 78, 046207 (2008)MathSciNet

82.

Panfilov, A.V., Hogeweg, P.: Scroll breakup in a three-dimensional excitable medium. Phys. Rev. E 53, 1740–1743 (1996)

83.

Henry, H., Hakim, V.: Linear stability of scroll waves. Phys. Rev. Lett. 85, 5328–5331 (2000)

84.

Luengviriya, C., Hauser, M.J.B.: Stability of scroll ring orientation in an advective field. Phys. Rev. E 77, 056214 (2008)

85.

Henry, H., Hakim, V.: Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states. Phys. Rev. E 65, 046235 (2002)MathSciNet

86.

Henry, H.: Spiral wave drift in an electric field and scroll wave instabilities. Phys. Rev. E 70, 026204 (2004)

87.

Wang, C., Wang, S., Zhang, C., Ouyang, Q.: Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients. Phys. Rev. E 72, 066207 (2005)

88.

Luengviriya, C., Storb, U., Lindner, G., et al.: Scroll wave instabilities in an excitable chemical medium. Phys. Rev. Lett. 100, 148302 (2008)

89.

Aranson, I., Mitkov, I.: Helicoidal instability of a scroll vortex in three-dimensional reaction–diffusion systems. Phys. Rev. E 58, 4556–4559 (1998)

90.

Qu, Z., Xie, F., Garfinkel, A.: Diffusion-induced vortex filament instability in 3-dimensional excitable media. Phys. Rev. Lett. 83, 2668–2671 (1999)

91.

Aranson, I.S., Bishop, A.R., Kramer, L.: Dynamics of vortex lines in the three-dimensional complex Ginzburg–Landau equation: instability, stretching, entanglement, and helices. Phys. Rev. E 57, 5276–5286 (1998)

92.

Nam, K., Ott, E., Guzdar, P.N., Gabbay, M.: Stability of spiral wave vortex filament with phase twists. Phys. Rev. E 58, 2580–2585 (1998)MathSciNet

93.

Wellner, M., Berenfeld, O., Pertsov, A.M.: Predicting filament drift in twisted anisotropy. Phys. Rev. E 61, 1845–1850 (2000)

94.

Setayeshgar, S., Bernoff, A.J.: Scroll waves in the presence of slowly varying anisotropy with application to the heart. Phys. Rev. Lett. 88, 028101 (2002)

95.

Verschelde, H., Dierckx, H., Bernus, O.: Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue. Phys. Rev. Lett. 99, 168104 (2007)

96.

Vinson, M., Pertsov, A.: Dynamics of scroll rings in a parameter gradient. Phys. Rev. E 59, 2764–2771 (1999)MathSciNet

97.

Gabbay, M., Ott, E., Guzdar, P.N.: The dynamics of scroll wave filaments in the complex Ginzburg–Landau equation. Phys. D 118, 371–395 (1998)MATHMathSciNet

98.

ten Tusscher, K.H.W.J., Panfilov, A.V.: Eikonal formulation of the minimal principle for scroll wave filaments. Phys. Rev. Lett. 93, 108106 (2004)

99.

Dierckx, H., Bernus, O., Verschelde, H.: A geometric theory for scroll wave filaments in anisotropic excitable media. Phys. D 238, 941–950 (2009)MATHMathSciNet

100.

Alonso, S., Panfilov, A.V.: Negative filament tension at high excitability in a model of cardiac tissue. Phys. Rev. Lett. 100, 218101 (2008)

101.

Gabbay, M., Ott, E., Guzdar, P.N.: Motion of scroll wave filaments in the complex Ginzburg–Landau equation. Phys. Rev. Lett. 78, 2012–2015 (1997)

102.

Alonso, S., Kähler, R., Mikhailov, A.S., Sagués, F.: Expanding scroll rings and negative tension turbulence in a model of excitable media. Phys. Rev. E 70, 056201 (2004)MathSciNet

103.

Luengviriya, C., Müller, S.C., Hauser, M.J.B.: Reorientation of scroll rings in an advective field. Phys. Rev. E 77, 015201(R) (2008)

104.

Bray, M.A., Wikswo, J.P.: Interaction dynamics of a pair of vortex filament rings. Phys. Rev. Lett. 90, 238303 (2003)

105.

Bánsági, T., Steinbock, O.: Nucleation and collapse of scroll rings in excitable media. Phys. Rev. Lett. 97, 198301 (2006)

106.

Wu, Y., Qiao, C., Ouyang, Q., Wang, H.: Control of spiral turbulence by periodic forcing in a reaction–diffusion system with gradients. Phys. Rev. E 77, 036226 (2008)

107.

Alonso, S., Sancho, J.M., Sagués, F.: Suppression of scroll wave turbulence by noise. Phys. Rev. E 70, 067201 (2004)

108.

Wu, N.J., Zhang, H., Ying, H.P., et al.: Suppression of winfree turbulence under weak spatiotemporal perturbation. Phys. Rev. E 73, 060901(R) (2006)

109.

Briscolini, M., Santangelo, P., Succi, S., Benzi, R.: Extended self-similarity in the numerical simulation of three-dimensional homogeneous flows. Phys. Rev. E 50, R1745–R1747 (1994)

110.

Barkley, D.: A model for fast computer simulation of waves in excitable media. Phys. D 49, 61–70 (1991)

111.

Rusakov, A., Medvinsky, A.B., Panfilov, A.V.: Scroll waves meandering in a model of an excitable medium. Phys. Rev. E 72, 022902 (2005)MathSciNet

112.

Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, Cambridge (1970)

Titel: Three-Dimensional Lattice Boltzmann Model for the Complex Ginzburg–Landau Equation
verfasst von: Jianying Zhang
Guangwu Yan
Publikationsdatum: 01.09.2014
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2014
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-013-9811-z

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