September  2020, 28(3): 1207-1225. doi: 10.3934/era.2020066

A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

2. 

School of Mathematics and Computational Science, Hunan First Normal University, Changsha, China

* Corresponding author: Xuguang Yang

Received  April 2020 Revised  June 2020 Published  July 2020

Fund Project: Z. Qiao's work is partially supported by Hong Kong Research Council GRF grant 15300417. X. Yang's work is supported by the Natural Science Foundation of China (Grant No. 11802090) and was also supported by the Hong Kong Polytechnic University Postdoctoral Fellowships Scheme (No. 1-YW1D)

In this work, a multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is proposed to solve a coupled chemotaxis-fluid model. In the evolution equation of the proposed LBM, Beam-Warming (B-W) scheme is used to enhance the numerical stability. In numerical experiments, at first, the comparison between the classical LBM and the present LBM with B-W scheme is carried out by simulating blow up phenomenon of the Keller-Segel (K-S) model. Numerical results show that the stability of the present LBM with B-W scheme is better than the classical one. Then, the second order convergence rate of the proposed B-W scheme is verified in the numerical study of the coupled Navier-Stokes (N-S) K-S model. Finally, through solving the coupled chemotaxis-fluid model, the formation of falling bacterial plumes is numerically investigated. Numerical results agree well with existing ones in the literature.

Citation: Zhonghua Qiao, Xuguang Yang. A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28 (3) : 1207-1225. doi: 10.3934/era.2020066
References:
[1]

M. BurgerM. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.  doi: 10.1137/050637923.  Google Scholar

[2]

Z. Chai, H. Liang, R. Du and B. Shi, A lattice Boltzmann model for two-phase flow in porous media, SIAM J. Sci. Comput., 41 (2019), B746–B772. doi: 10.1137/18M1166742.  Google Scholar

[3]

Z. H. Chai and T. S. Zhao, Lattice Boltzmann model for the convection-diffusion equation, Phy. Rev. E, 87 (2013), 063309. Google Scholar

[4]

S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30 (1998), 329-364.  doi: 10.1146/annurev.fluid.30.1.329.  Google Scholar

[5]

A. ChertockK. FellnerA. KurganovA. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.  doi: 10.1017/jfm.2011.534.  Google Scholar

[6]

Y. DeleuzeC.-Y. ChiangM. Thiriet and Tony W. H. Sheu, Numerical study of plume patterns in a chemotaxis-diffusion-convection coupling system, Comput. Fluids, 126 (2016), 58-70.  doi: 10.1016/j.compfluid.2015.10.018.  Google Scholar

[7]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308.  doi: 10.1137/040612841.  Google Scholar

[8]

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103. doi: 10.1103/PhysRevLett.93.098103.  Google Scholar

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Y. Epshteyn, Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model, J. Sci. Comput., 53 (2012), 689-713.  doi: 10.1007/s10915-012-9599-2.  Google Scholar

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Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxisn models, J. Comput. Appl. Math., 224 (2009), 168-181.  doi: 10.1016/j.cam.2008.04.030.  Google Scholar

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Y. Epshteyn and A. Kurganov, New interior penalyt discontinous Galerkin methods for the Keller-Segel chemotaxis model, SIAM J. Numer. Anal., 47 (2008/2009), 386-408.  doi: 10.1137/07070423X.  Google Scholar

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F. Filbet, A finit volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.  doi: 10.1007/s00211-006-0024-3.  Google Scholar

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N. A. Hill and T. J. Pedley, Bioconvection, Fluid Dynam. Res., 37 (2005), 1-20.  doi: 10.1016/j.fluiddyn.2005.03.002.  Google Scholar

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M. Hilpert, Lattice-Boltzmann model for bacterial chemotaxis, J. Math. Biol., 51 (2005), 302-332.  doi: 10.1007/s00285-005-0318-6.  Google Scholar

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H. G. Lee and J. Kim, Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber, Eur. J. Mech. B Fluid., 52 (2015), 120-130.  doi: 10.1016/j.euromechflu.2015.03.002.  Google Scholar

[18]

Y. H. QianS. Succi and S. A. Orazag, Recent advances in lattice Boltzmann computing., Annu. Rev. Comput. Phys., 3 (1995), 195-242.   Google Scholar

[19]

Z. QiaoX. Yang and Y. Zhang, Thermodynamic-consistent multiple-relaxation-time lattice Boltzmann equation model for two-phase hydrocarbon fluids with Peng-Robinson equation of state, Int. J. Heat Mass Tran., 141 (2019), 1216-1226.  doi: 10.1016/j.ijheatmasstransfer.2019.07.023.  Google Scholar

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D. L. Ropp and J. N. Shadid, Stability of operator splitting methods for systems with indefinite poerators: advection-diffusion-reaction systems, J. Comput. Phys., 228 (2009), 3508-3516.  doi: 10.1016/j.jcp.2009.02.001.  Google Scholar

[21]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modeling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.  doi: 10.1093/imanum/drl018.  Google Scholar

[22]

T. W. H. Sheu and C. Y. Chiang, Numerical investigation of chemotaxic phenomenon in incompressible viscous fluid flow, Comput. Fluids, 103 (2014), 290-306.  doi: 10.1016/j.compfluid.2014.07.023.  Google Scholar

[23]

B. C. Shi and Z. L. Guo, Lattice Boltzmann model for nonlinear conveciton-diffusion equations, Phy. Rev. E, 79 (2009), 016701. Google Scholar

[24]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci., 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[25]

R. TysonL. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), 455-475.  doi: 10.1007/s002850000038.  Google Scholar

[26]

L. WangY. ZhaoX. YangB. Shi and Z. Chai, A lattice Boltzmann analysis of the conjugate natural convection in square enclosure with a circular cylinder, Appl. Math. Model., 71 (2019), 31-44.  doi: 10.1016/j.apm.2019.02.012.  Google Scholar

[27]

Z. F. Yan and M. Hilpert, A multiple-relaxation-time Lattice-Boltzmann model for bacterial chemotaxis: Effects of initial concentration, diffusion, and hydrodynamic dispersion on traveling bacterial bands, Bull Math. Biol., 76 (2014), 2449-2475.  doi: 10.1007/s11538-014-0020-1.  Google Scholar

[28]

X. YangB. ShiZ. Chai and Z. Guo, A coupled lattice Boltzmann method to solve Nernst-Planck model for simulating electro-osmotic flows, J. Sci. Comput., 61 (2014), 222-238.  doi: 10.1007/s10915-014-9820-6.  Google Scholar

[29]

T. Zhang, B. C. Shi, Z. L. Guo, Z. H. Chai and J. H. Lu, Genearl bounce-back scheme for concentration boundary condition in the lattice Boltzmann method, Phys. Rev. E, 85 (2012), 016701. Google Scholar

show all references

References:
[1]

M. BurgerM. Di Francesco and Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.  doi: 10.1137/050637923.  Google Scholar

[2]

Z. Chai, H. Liang, R. Du and B. Shi, A lattice Boltzmann model for two-phase flow in porous media, SIAM J. Sci. Comput., 41 (2019), B746–B772. doi: 10.1137/18M1166742.  Google Scholar

[3]

Z. H. Chai and T. S. Zhao, Lattice Boltzmann model for the convection-diffusion equation, Phy. Rev. E, 87 (2013), 063309. Google Scholar

[4]

S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30 (1998), 329-364.  doi: 10.1146/annurev.fluid.30.1.329.  Google Scholar

[5]

A. ChertockK. FellnerA. KurganovA. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.  doi: 10.1017/jfm.2011.534.  Google Scholar

[6]

Y. DeleuzeC.-Y. ChiangM. Thiriet and Tony W. H. Sheu, Numerical study of plume patterns in a chemotaxis-diffusion-convection coupling system, Comput. Fluids, 126 (2016), 58-70.  doi: 10.1016/j.compfluid.2015.10.018.  Google Scholar

[7]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308.  doi: 10.1137/040612841.  Google Scholar

[8]

C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., 93 (2004), 098103. doi: 10.1103/PhysRevLett.93.098103.  Google Scholar

[9] M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.   Google Scholar
[10]

Y. Epshteyn, Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model, J. Sci. Comput., 53 (2012), 689-713.  doi: 10.1007/s10915-012-9599-2.  Google Scholar

[11]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxisn models, J. Comput. Appl. Math., 224 (2009), 168-181.  doi: 10.1016/j.cam.2008.04.030.  Google Scholar

[12]

Y. Epshteyn and A. Kurganov, New interior penalyt discontinous Galerkin methods for the Keller-Segel chemotaxis model, SIAM J. Numer. Anal., 47 (2008/2009), 386-408.  doi: 10.1137/07070423X.  Google Scholar

[13]

F. Filbet, A finit volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.  doi: 10.1007/s00211-006-0024-3.  Google Scholar

[14]

N. A. Hill and T. J. Pedley, Bioconvection, Fluid Dynam. Res., 37 (2005), 1-20.  doi: 10.1016/j.fluiddyn.2005.03.002.  Google Scholar

[15]

A. J. Hillesdon and T. J. Pedley, Bioconvection in suspensions of oxytactic bacteria: Linear theory, J. Fluid Mech., 324 (1996), 223-259.  doi: 10.1017/S0022112096007902.  Google Scholar

[16]

M. Hilpert, Lattice-Boltzmann model for bacterial chemotaxis, J. Math. Biol., 51 (2005), 302-332.  doi: 10.1007/s00285-005-0318-6.  Google Scholar

[17]

H. G. Lee and J. Kim, Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber, Eur. J. Mech. B Fluid., 52 (2015), 120-130.  doi: 10.1016/j.euromechflu.2015.03.002.  Google Scholar

[18]

Y. H. QianS. Succi and S. A. Orazag, Recent advances in lattice Boltzmann computing., Annu. Rev. Comput. Phys., 3 (1995), 195-242.   Google Scholar

[19]

Z. QiaoX. Yang and Y. Zhang, Thermodynamic-consistent multiple-relaxation-time lattice Boltzmann equation model for two-phase hydrocarbon fluids with Peng-Robinson equation of state, Int. J. Heat Mass Tran., 141 (2019), 1216-1226.  doi: 10.1016/j.ijheatmasstransfer.2019.07.023.  Google Scholar

[20]

D. L. Ropp and J. N. Shadid, Stability of operator splitting methods for systems with indefinite poerators: advection-diffusion-reaction systems, J. Comput. Phys., 228 (2009), 3508-3516.  doi: 10.1016/j.jcp.2009.02.001.  Google Scholar

[21]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modeling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.  doi: 10.1093/imanum/drl018.  Google Scholar

[22]

T. W. H. Sheu and C. Y. Chiang, Numerical investigation of chemotaxic phenomenon in incompressible viscous fluid flow, Comput. Fluids, 103 (2014), 290-306.  doi: 10.1016/j.compfluid.2014.07.023.  Google Scholar

[23]

B. C. Shi and Z. L. Guo, Lattice Boltzmann model for nonlinear conveciton-diffusion equations, Phy. Rev. E, 79 (2009), 016701. Google Scholar

[24]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci., 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.  Google Scholar

[25]

R. TysonL. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), 455-475.  doi: 10.1007/s002850000038.  Google Scholar

[26]

L. WangY. ZhaoX. YangB. Shi and Z. Chai, A lattice Boltzmann analysis of the conjugate natural convection in square enclosure with a circular cylinder, Appl. Math. Model., 71 (2019), 31-44.  doi: 10.1016/j.apm.2019.02.012.  Google Scholar

[27]

Z. F. Yan and M. Hilpert, A multiple-relaxation-time Lattice-Boltzmann model for bacterial chemotaxis: Effects of initial concentration, diffusion, and hydrodynamic dispersion on traveling bacterial bands, Bull Math. Biol., 76 (2014), 2449-2475.  doi: 10.1007/s11538-014-0020-1.  Google Scholar

[28]

X. YangB. ShiZ. Chai and Z. Guo, A coupled lattice Boltzmann method to solve Nernst-Planck model for simulating electro-osmotic flows, J. Sci. Comput., 61 (2014), 222-238.  doi: 10.1007/s10915-014-9820-6.  Google Scholar

[29]

T. Zhang, B. C. Shi, Z. L. Guo, Z. H. Chai and J. H. Lu, Genearl bounce-back scheme for concentration boundary condition in the lattice Boltzmann method, Phys. Rev. E, 85 (2012), 016701. Google Scholar

Figure 1.  Numerical solutions of bacteria concentrations $ n(\mathbf{x}, t) $ of (27)-(28) by the standard LBM (A = 0): (a) $ t = 5.0\times10^{-6} $, (b) $ t = 1.0\times10^{-5} $, (c) $ t = 4.5\times10^{-5} $, (d) $ t = 1.15\times10^{-4} $
Figure 2.  Numerical solutions of bacteria concentrations $ n(\mathbf{x}, t) $ of (27)-(28) by the present LBM with B-W scheme (A = 0.5), (a) $ t = 5.0\times10^{-6} $, (b) $ t = 1.0\times10^{-5} $, (c) $ t = 4.5\times10^{-5} $, (d) $ t = 1.15\times10^{-4} $
Figure 3.  1D profiles of $ n(\mathbf{x}, t) $ along y = 0 at $ t = 1.15\times10^{-4} $: (a) by the standard LBM (A = 0); (b) by the present LBM with B-W scheme (A = 0.5)
Figure 4.  (a) Vertical profiles of the oxygen $ c $ and cell densities $ n $ and (b) streamlines at t = 0.22
Figure 5.  Vertical profiles of (a) oxygen $ c $ and (b) cell densities $ n $ at t = 0.22 with $ \alpha = 5 $, $ \delta = 1 $ and different $ \beta $
Figure 6.  Vertical profiles of (a) oxygen $ c $ and (b) cell densities $ n $ at t = 0.22 with $ \beta = 10 $, $ \delta = 1 $ and different $ \alpha $
Figure 7.  Bioconvection patterns of the cell density $ n(x, y) $ at (a) $ t = 0.32 $, (b) $ t = 0.37 $, (c) $ t = 0.42 $, and (d) $ t = 0.47 $
Figure 8.  For $ \delta = 5 $, cell concentration $ n(x, y) $ at (a) $ t = 0.27 $, (b) $ t = 0.34 $, and (c) $ t = 0.39 $
Figure 9.  For $ \delta = 25 $, cell concentration $ n(x, y) $ at (a) $ t = 2.9 $, (b) $ t = 3.4 $, and (c) $ t = 3.9 $
Figure 10.  For $ \delta = 50 $, cell concentration $ n(x, y) $ at (a) $ t = 1.9 $, (b) $ t = 2.4 $, and (c) $ t = 2.9 $
Table 1.  Convergence test with different lattice spacings, $ \delta t = 1.0\times10^{-5} $
$ Meshes $ $ E_n $ order $ E_c $ order $ E_\mathbf{u} $ order
$ 64\times 64 $ $ 1.3934 \times 10^{-4} $ $ 1.6327 \times 10^{-4} $ $ 2.0029 \times 10^{-4} $
$ 128\times 128 $ $ 3.6892 \times 10^{-5} $ 1.9172 $ 4.3355 \times10^{-5} $ 1.9130 $ 5.3150 \times 10^{-5} $ 1.9139
$ 256\times 256 $ $ 9.4955 \times 10^{-6} $ 1.9580 $ 1.1174\times 10^{-5} $ 1.9560 $ 1.3699 \times 10^{-5} $ 1.9560
$ 512\times 512 $ $ 2.4090 \times 10^{-6} $ 1.9788 $ 2.8380\times 10^{-6} $ 1.9772 $ 3.4775 \times 10^{-6} $ 1.9779
$ Meshes $ $ E_n $ order $ E_c $ order $ E_\mathbf{u} $ order
$ 64\times 64 $ $ 1.3934 \times 10^{-4} $ $ 1.6327 \times 10^{-4} $ $ 2.0029 \times 10^{-4} $
$ 128\times 128 $ $ 3.6892 \times 10^{-5} $ 1.9172 $ 4.3355 \times10^{-5} $ 1.9130 $ 5.3150 \times 10^{-5} $ 1.9139
$ 256\times 256 $ $ 9.4955 \times 10^{-6} $ 1.9580 $ 1.1174\times 10^{-5} $ 1.9560 $ 1.3699 \times 10^{-5} $ 1.9560
$ 512\times 512 $ $ 2.4090 \times 10^{-6} $ 1.9788 $ 2.8380\times 10^{-6} $ 1.9772 $ 3.4775 \times 10^{-6} $ 1.9779
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