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Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations. (English) Zbl 1411.76146
Summary: In this paper, a finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed. In this model, the equilibrium distribution function is delicately designed in order to recover the convection-diffusion equation exactly. Different from the standard LB model, the temporal and spatial steps in this model are decoupled such that it is convenient to study convection-diffusion problem with the non-uniform grid. In addition, it also preserves the advantage of standard LB model that the complex-valued convection-diffusion equation can be solved directly. The von Neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model. To test the performance of the model, a series of numerical simulations of some classical problems, including the diffusion equation, the nonlinear heat conduction equation, the Sine-Gordon equation, the Gaussian hill problem, the Burgers-Fisher equation, and the nonlinear Schrödinger equation, have also been carried out. The results show that the present model has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.

##### MSC:
 76M28 Particle methods and lattice-gas methods 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics 65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations
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##### References:
 [1] Cussler, E. L., Diffusion: mass transfer in fluid systems, (2009), Cambridge University Press New York [2] Thomas, J. W., Numerical partial differential equations: finite difference methods, (2013), Springer Science & Business Media New York [3] Johnson, C., Numerical solution of partial differential equations by the finite element method, (2012), Courier Corporation New York [4] LeVeque, R. J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040 [5] Guo, Z.; Shu, C., Lattice Boltzmann method and its applications in engineering, (2013), World Scientific Singapore · Zbl 1278.76001 [6] Liang, H.; Shi, B. C.; Chai, Z. H., Lattice Boltzmann modeling of three-phase incompressible flows, Phys. Rev. E, 93, 1, 013308, (2016) [7] Zheng, H. W.; Shu, C.; Chew, Y. T., A lattice Boltzmann model for multiphase flows with large density ratio, J. Comput. Phys., 218, 1, 353-371, (2006) · Zbl 1158.76419 [8] Chen, L.; Kang, Q.; Mu, Y., A critical review of the pseudopotential multiphase lattice Boltzmann model: methods and applications, Int. J. Heat Mass Transf., 76, 210-236, (2014) [9] Chai, Z.; Huang, C.; Shi, B., A comparative study on the lattice Boltzmann models for predicting effective diffusivity of porous media, Int. J. Heat Mass Transf., 98, 687-696, (2016) [10] Guo, Z.; Zhao, T., Lattice Boltzmann model for incompressible flows through porous media, Phys. Rev. E, 66, 3, 036304, (2002) [11] Dou, Z.; Zhou, Z. F., Numerical study of non-uniqueness of the factors influencing relative permeability in heterogeneous porous media by lattice Boltzmann method, Int. J. Heat Fluid Flow, 42, 23-32, (2013) [12] Huang, C.; Chai, Z.; Shi, B., Non-Newtonian effect on hemodynamic characteristics of blood flow in stented cerebral aneurysm, Comput. Phys., 13, 916-928, (2013) [13] Dawson, S. P.; Chen, S.; Doolen, G. D., Lattice Boltzmann computations for reaction-diffusion equations, J. Chem. Phys., 98, 2, 1514-1523, (1993) [14] Guo, Z.; Shi, B.; Wang, N. C., Fully Lagrangian and lattice Boltzmann methods for the advection-diffusion equation, J. Sci. Comput., 14, 3, 291-300, (1999) · Zbl 0971.76073 [15] Van der Sman, R. G.M.; Ernst, M. H., Convection-diffusion lattice Boltzmann scheme for irregular lattices, J. Comput. Phys., 160, 2, 766-782, (2000) · Zbl 1040.76514 [16] Yu, X.; Shi, B., A lattice Boltzmann model for reaction dynamical systems with time delay, Appl. Math. Comput., 181, 2, 958-965, (2006) · Zbl 1204.76030 [17] Shi, B.; Deng, B.; Du, R., A new scheme for source term in LBGK model for convection-diffusion equation, Comput. Math. Appl., 55, 7, 1568-1575, (2008) · Zbl 1142.76477 [18] Chai, Z. H.; Shi, B., A novel lattice Boltzmann model for the Poisson equation, Appl. Math. Model., 32, 10, 2050-2058, (2008) · Zbl 1145.82344 [19] Xiang, X.; Wang, Z.; Shi, B., Modified lattice Boltzmann scheme for nonlinear convection diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 17, 6, 2415-2425, (2012) · Zbl 1335.76044 [20] Chai, Z.; Zhao, T., Lattice Boltzmann model for the convection-diffusion equation, Phys. Rev. E, 87, 6, 063309, (2013) [21] Zhang, X.; Bengough, A. G.; Crawford, J. W., A lattice BGK model for advection and anisotropic dispersion equation, Adv. Water Resour., 25, 1, 1-8, (2002) [22] Ginzburg, I., Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv. Water Resour., 28, 11, 1171-1195, (2005) [23] Yoshida, H.; Nagaoka, M., Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation, J. Comput. Phys, 229, 20, 7774-7795, (2010) · Zbl 1425.76204 [24] Chopard, B.; Falcone, J. L.; Latt, J., The lattice Boltzmann advection-diffusion model revisited, Eur. Phys. J. Spec. Top, 171, 1, 245-249, (2009) [25] Shi, B.; Guo, Z., Lattice Boltzmann model for nonlinear convection-diffusion equations, Phys. Rev. E, 79, 1, 016701, (2009) [26] Chai, Z.; Shi, B.; Guo, Z., A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations, J. Sci. Comput., 69, 1, 355-390, (2016) · Zbl 1356.82032 [27] He, X.; Doolen, G., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134, 2, 306-315, (1997) · Zbl 0886.76072 [28] He, X.; Doolen, G. D., Lattice Boltzmann method on a curvilinear coordinate system: vortex shedding behind a circular cylinder, Phys. Rev. E, 56, 1, 434, (1997) [29] Lallemand, P.; Luo, L. S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61, 6, 6546, (2000) [30] Lu, J. H.; Chai, Z. H.; Shi, B. C., Rectangular lattice Boltzmann model for nonlinear convection-diffusion equations, Phil. Trans. R. Soc. A, 369, 1944, 2311-2319, (2011) · Zbl 1223.76084 [31] Guo, Z.; Zhao, T., Explicit finite-difference lattice Boltzmann method for curvilinear coordinates, Phys. Rev. E, 67, 6, 066709, (2003) [32] Cao, N.; Chen, S.; Jin, S., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Phys. Rev. E, 55, 1, R21, (1997) [33] Mei, R.; Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. Comput. Phys., 143, 2, 426-448, (1998) · Zbl 0934.76074 [34] Bardow, A.; Karlin, I. V.; Gusev, A. A., General characteristic-based algorithm for off-lattice Boltzmann simulations, Europhys. Lett, 75, 3, 434, (2006) [35] Sterling, J. D.; Chen, S., Stability analysis of lattice Boltzmann methods, J. Comput. Phys, 123, 1, 196-206, (1996) · Zbl 0840.76078 [36] Guo, Z.; Zheng, C.; Shi, B., Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chin. Phys., 11, 4, 366, (2002) [37] Sheng, Q.; Khaliq, A. Q.M.; Voss, D. A., Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Simul., 68, 4, 355-373, (2005) · Zbl 1073.65095 [38] Huang, R.; Wu, H., A modified multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation, J. Comput. Phys, 274, 50-63, (2014) · Zbl 1351.76237 [39] Wazwaz, A. M., The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations, Appl. Math. Comput., 169, 1, 321-338, (2005) · Zbl 1121.65359 [40] Xu, Y.; Shu, C. W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205, 1, 72-97, (2005) · Zbl 1072.65130
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