Abstract
A lattice Bhatnagar–Gross–Krook (LBGK) model for reaction-diffusion systems is presented. This model provides a mesoscopic approach to the dynamics of spatially-distributed reacting systems. Pure diffusion phenomena are computed and the results are agreement with the theoretical predictions. This method is also applied to formation of Turing patterns in chloride-iodide-malonic acid (CIMA) reactive model. We get hexagonal structures and stripes which are agreement with other numerical results and experimental results.
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REFERENCES
Turing, A. M. (1952). The chemical basis of morphogenesis. Philos.Trans.Roy.Soc.London Ser.B 237, 37.
De Kepper, P., Castets, V., and Dulos, E. (1991). Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction. Physica D 49, 161–169.
Ouyang, Q., and Swinney, H. L. (1991). Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612.
Lengyel, I., and Epstein, I. R. (1991). Modeling of Turing structures in the chlorite-iodidemalonic acid-starch reaction system. Science 251, 650–652.
Jensen, O., Pannbacker, V. O., and Dewel, G., et al. (1990). Subcritical transitions to Turing structures. Phys.Lett.A 179, 91–96.
Rudovics, B., Barillot, E., and Davies, P. W. (1999). Experimental studies and quantitative modeling of Turing patterns in the (chlorine dioxide, iodine, malonic acid) reaction. J.Chem.Phys.A 103, 1790–1800.
Markus, M., and Hess, B. (1990). Isotropic cellular automaton for modelling excitable media. Nature 347, 56–58.
Gerhardt, M., Schuster, H., and Tyson, J. (1990). A cellular automaton model of excitable media including curvature and dispersion. Science 247, 1563–1566.
Schepers, H. E., and Markus, M. (1992). Two types of performance of an isotropic cellular automaton: stationary (Turing) patterns and spiral waves. Physica A 188, 337–343.
Weimar, J. R., and Boon, J. P. (1994). New class of cellular automata for reaction-diffusion systems applied to the CIMA reaction. In Lawniczak, L., and Kapral, R. (eds.), Lattice Gas Automata and Pattern Formation, Waterloo, Ontario, Canada, Fields Institute.
Boon, J. P., Dab, D., and Kapral, R. (1996). Lattice gas automata for reactive systems. Phys.Rep. 237, 55–147.
Benzi, R., Succi, S., and Vergassola, M. (1992). The lattice Boltzmann equation: Theroy and applications. Phys.Rep. 222(3), 145–197.
Chen, H., Chen, S., and Matthaeus, W. (1992). Recovery of the Navier-Stokes equations using a lattice gas Boltzmann method. Phys.Rev.A 45, 5339–5342.
Qian, Y. H., d'Humières, D., and Lallemand, P. (1992). Lattice BGK models for Navier-Stokes equation. Europhys.Lett. 17, 479–484.
Qian, Y. H., Succi, S., and Orszag, S. A. (1995). Recent advances in lattice Boltzmann computing. Ann.Rev.Comp.Phys. 3, 195–242.
Chen, S., and Doolen, G. D. (1998). Lattice Boltzmann method for fluid flows. Ann.Rev.Fluid Mech. 30, 329–364.
Dawson, S. P., Chen, S., and Doolen, G. D. (1993). Lattice Boltzmann computations for reaction-diffusion equations. J.Chem.Phys. 98(2), 1514–1523.
Wolf-Gladrow, D. (1995). A lattice Boltzmann equation for diffusion. J.Stat.Phys. 79, 1023–1032.
Wolf-Gladrow, D. (2000). Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer-Verlag, Berlin/Heidelberg.
Blaak, R., and Sloot, P. M. (2000). Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. Comput.Phys.Comm. 129, 256–266.
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Li, Q., Zheng, CG., Wang, NC. et al. LBGK Simulations of Turing Patterns in CIMA Model. Journal of Scientific Computing 16, 121–134 (2001). https://doi.org/10.1023/A:1012278606077
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DOI: https://doi.org/10.1023/A:1012278606077