Abstract
In this paper, we first deduce an improved chemotaxis-fluid system by introducing a chemotactic stress force, which can be used to describe the chemotactic movement of bacteria in a viscous fluid. Compared with the classical chemotaxis-Navier–Stokes system, the newly modified system obeys the law of energy dissipation. To solve such a chemotaxis-fluid system, we develop a linear, decoupled fully-discrete finite element scheme by combining the scalar auxiliary variable (SAV) approach, implicit-explicit (IMEX) scheme and pressure-projection method. The unconditional energy stability of the developed scheme is proved rigorously, and we further prove the optimal error estimates for the fully discrete scheme, especially for the pressure bound. Finally, some numerical examples are presented to verify the accuracy, energy stability and performance of the proposed numerical scheme.
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References
Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Model Meth. Appl. Sci. 25(09), 1663–1763 (2015)
Barrett, J.W., Blowey, J.F.: Finite element approximation of a nonlinear cross-diffusion population model. Numer. Math. 98(2), 195–221 (2004)
Chertock, A., Fellner, K., Korganov, A., Lorz, A., Markowich, P.: Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach. J. Fluid Mech. 694, 155–190 (2012)
Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to coupled chemotaxis-fluid equations. Discr. Cont. Dyn. Syst. A 33(6), 2271–2297 (2013)
Ciarlet, P.G.: Introduction to linear shell theory, Series in applied mathematics. Elsevier, Armsterdam (1998)
Cai, X., Jiu, Q.: Weak and strong solutions for the incompressible Navier–Stokes equations with damping. J. Math. Anal. Appl. 343, 799–809 (2008)
Choi, H., Shen, J.: Efficient splitting schemes for magneto-hydrodynamic equations. Sci. China Math. 59(8), 1495–1510 (2016)
Deleuze, Y., Chiang, C.Y., Thiriet, M., Sheu, T.W.: Numerical study of plume patterns in a chemotaxis-diffusion-convection coupling system. Comput. Fluids 126, 58–70 (2016)
Duan, R., Lorz, A., Markowich, P.A.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Part. Differ. Equ. 35(9), 1635–1673 (2010)
Duarte-Rodríguez, A., Rodríguez-Bellido, M.A., Rueda-Gómez, D.A., Villamizar-Roa E.J.: Numerical analysis for a chemotaxis-Navier–Stokes system, ESAIM:M2AN. 55, S417–S445 (2021)
Epshteyn, Y., Izmirliouglu, A.: Fully discrete analysis of a discontinuous finite element method for the Keller–Segel chemotaxis model. J. Sci. Comput. 40(1–3), 211–256 (2009)
Elliott, C.M.: Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7(1), 61–71 (1987)
Girault, V., Raviart, P.A.: Finite Element Method for Navier–Stokes equations: theory and Algorithms. Springer, Berlin (1987)
Guillén-González, F., Rodríguez-Bellido, M.A., Rueda-Gómez. D.A.: Unconditionally energy stable fully discrete schemes for a chemo-repulsion model, Math. Comp. 88 (319), 2069–2099 (2019)
Hillen, T., Painter, K.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1), 183–217 (2009)
Hillesdon, A.J., Pedley, T.J., Kessler, O.: The development of concentration gradients in a suspension of chemotactic bacteria. Bull. Math. Biol. 57(2), 299–344 (1995)
Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem, I: regularity of solutions and second order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)
Ivancic, F., Sheu, T.W.H., Solovchuk, M.: The free surface effect on a chemotaxis-diffusion-convection coupling system. Comput. Methods Appl. Mech. Eng. 356(1), 387–406 (2019)
Ivani, F., Sheu, T.W.H., Solovchuk, M.: Bacterial chemotaxis in thin fluid layers with free surface. Phys. Fluids. 32(6), 061902 (2020)
Jong, U.K.: Smooth solutions to a quasi-linear system of diffusion equations for a certain population model. Nonlin. Anal. 8(10), 1121–1144 (1984)
Kozono, H., Miura, M., Sugiyama, Y.: Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid. J. Funct. Anal. 270(5), 1663–1683 (2016)
Karimi, A., Paul, M.: Bioconvection in spatially extend domains. Phys. Rev. E. 87(5), 053016 (2013)
Kechkar, N., Silvester, D.: Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comp. 58(197), 1–11 (1992)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 339–415 (1970)
Kay, D., Silvester, D.: A posteriori error estimation for stabilized mixed approximations of the Stokes equations. SIAM J. Sci. Comput. 21(4), 1321–1336 (2000)
Lee, H.G., Kim, J.: Numerical investigation of falling bacterial plumes caused by bioconvection in a three-dimensional chamber. Eur. J. Mech. B/Fluids 52, 120–130 (2015)
Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179, 211–228 (2003)
Li, J., Li, R., Zhao, X., Chen, Z.: A second-order fractional time-stepping method for a coupled Stokes/Darcy system. J. Comput. Appl. Math. 390, 113329 (2021)
Qiu, C.X., He, X.M., Li, J., Lin, Y.P.: A domain decomposition method for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition. J. Comput. Phys. 411(15), 109400 (2020)
Li, J., Wang, X., Al Mahbub, M.A., Zheng, H.B., Chen, Z.X.: Local and parallel efficient BDF2 and BDF3 rotational pressure-correction schemes for a coupled Stokes/Darcy system. J. Comput. Appl. Math. 412(1), 114326 (2022)
Li, J., Yao, M., Al Mahbub, M.A., Zheng, H.B.: The efficient rotational pressure-correction schemes for the coupling Stokes/Darcy Problem. Comput. Math. Appl. 79(2), 337–353 (2020)
Marrocco, A.: Numerical simulation of chemotactic bacteria aggregation via mixed finite elements. ESAIM:M2AN. 37(4), 617–630 (2003)
Suphantharika, M., Ison, A.P., Lilly, M.D., Buckland, B.C.: The influence of dissolved oxygen tension on the synthesis of the antibiotic difficidin by bacillus subtilis. Biotechnol. Bioeng. 44, 1007–1012 (1994)
Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)
Shen, J., Xu, J.: Unconditionally bound preserving and energy dissipative schemes for a class of Keller–Segel equations. SIAM J. Numer. Anal. 58(3), 1674–1695 (2020)
Saito, N.: Conservative upwind finite-element method for a simplified Keller–Segel system modeling chemotaxis. IMA J. Numer. Anal. 27(2), 332–365 (2007)
Strehl, R., Sokolov, A., Kuzmin, D., Horstmann, D., Turek, S.: A positivity-preserving finite element method for chemotaxis problems in 3D. J. Comput. Appl. Math. 239, 290–303 (2013)
Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O., Goldstein, R.E.: Bacterial swimming and oxygen transport near contact lines. Proc. Nat. Acad. Sci. USA 102(7), 2277–2282 (2005)
Winkler, M.: Global large-data solutions in a chemotaxis-(Navier–Stokes) system modelling cellular swimming in fluid drops. Commun. Part. Differ. Equ. 37(2), 319–351 (2012)
Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. Anal. 211(2), 455–487 (2014)
Wang, C., Wang, J., Xia, Z., Xu, L.: Optimal error estimates of a Crank–Nicolson finite element projection method for magnetohydrodynamic equations. ESAIM:M2AN. 56, 767–789 (2022)
Wang, X., Zou, G., Wang, B.: The stabilized penalty-projection finite element method for the Navier–Stokes–Cahn–Hilliard–Oono system. Appl. Numer. Math. 165, 376–413 (2021)
Yagi, A.: Global solution to some quasilinear parabolic system in population dynamics. Nonlin. Anal. 21(8), 603–630 (1993)
Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)
Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Methods Appl. Mech. Eng. 318, 1005–1029 (2017)
Yang, X., Zhang, G.D., He, X.M.: Convergence analysis of an unconditionally energy stable projection scheme for magneto-hydrodynamic equations. Appl. Numer. Math. 36, 235–256 (2019)
Zhang, Q., Zheng, X.: Global well-posedness for the two-dimensional incompressible chemotaxis-Navier–Stokes equations. SIAM. J. Math. Anal. 46(4), 3078–3105 (2014)
Zou, G., Wang, B., Yang, X.: A fully-decoupled discontinuous Galerkin approximation of the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth model. ESAIM:M2AN. 56, 2141–2180 (2022)
Acknowledgements
The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper. The work of Guang-an Zou is supported by China Postdoctoral Science Foundation (No.2019M662476), and the Key Scientific Research Projects of Colleges and Universities in Henan Province, China (23A110006). Jian Li is supported by NSF of China (No.11771259), Shaanxi Provincial Joint Laboratory of Artificial Intelligence (No.2022JC-SYS-05), Innovative team project of Shaanxi Provincial Department of Education(No.21JP013) and 2022 Shaanxi Provincial Social Science Fund Annual Project (No.2022D332).
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Tang, Y., Zou, Ga. & Li, J. Unconditionally Energy-Stable Finite Element Scheme for the Chemotaxis-Fluid System. J Sci Comput 95, 1 (2023). https://doi.org/10.1007/s10915-023-02118-4
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DOI: https://doi.org/10.1007/s10915-023-02118-4
Keywords
- Chemotaxis-fluid system
- Scalar auxiliary variable
- Finite element method
- Unconditionally energy-stable
- Error estimates
- Numerical examples