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Lifting in hybrid lattice Boltzmann and PDE models

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Computing and Visualization in Science

Abstract

Mathematical models based on kinetic equations are ubiquitous in the modeling of granular media, population dynamics of biological colonies, chemical reactions and many other scientific problems. These individual-based models are computationally very expensive because the evolution takes place in the phase space. Hybrid simulations can bring down this computational cost by replacing locally in the domain—in the regions where it is justified—the kinetic model with a more macroscopic description. This splits the computational domain into subdomains. The question is how to couple these models in a mathematically correct way with a lifting operator that maps the variables of the macroscopic partial differential equation to those of the kinetic model. Indeed, a kinetic model has typically more variables than a model based on a macroscopic partial differential equation and at each interface we need the missing data. In this contribution we report on different lifting operators for a hybrid simulation that combines a lattice Boltzmann model—a special discretization of the Boltzmann equation—with a diffusion partial differential equation. We focus on the numerical comparison of various lifting strategies.

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References

  1. Hundsdorfer W., Verwer J.G.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, Berlin (2003)

    MATH  Google Scholar 

  2. Davis T.: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia, PA (2006)

    Book  MATH  Google Scholar 

  3. Saad Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, PA (2003)

    Book  MATH  Google Scholar 

  4. Trottenberg U., Oosterlee C., Schüller A.: Multigrid. Academic Press, New York (2001)

    MATH  Google Scholar 

  5. Bellomo N., Delitala M., Coscia V.: On the mathematical theory of vehicular traffic flow, I, fluid dynamics and kinetic modelling. Math. Models Methods Appl. Sci. 12, 1801–1843 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Erban R., Othmer H.G.: From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math. 65, 361 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bittencourt J.A.: Fundamentals of Plasma Physics. Springer, New York (2004)

    MATH  Google Scholar 

  8. Flekkoy E.G., Wagner G., Feder J.: Hybrid model for combined particle and continuum dynamics. Europhys. Lett. 52(3), 271–276 (2000)

    Article  Google Scholar 

  9. Parks, M.L., Lehoucq, R.B.: Atomistic-to-Continuum Coupling. SIAM NEWS (2006)

  10. Garcia A.L., Bell J.B., Crutchfield W.Y., Alder B.J.: Adaptive mesh and algorithm refinement using direct simulation Monte Carlo. J. Comput. Phys. 154, 134–155 (1999)

    Article  MATH  Google Scholar 

  11. Dupuis A., Kotsalis E.M., Koumoutsakos P.: Coupling lattice Boltzmann and molecular dynamics models for dense fluids. Phys. Rev. E75, 046704 (2007)

    Google Scholar 

  12. Van Leemput P., Vandekerckhove C., Vanroose W., Roose D.: Accuracy of hybrid lattice Boltzmann/finite difference schemes for reaction-diffusion systems. Multiscale Model. Simul. 6, 838–857 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Samaey G., Vandekerckhove C., Vanroose W.: A multilevel algorithm to compute steady states of lattice Boltzmann Models. Coping with complexity: Model reduction and data analysis, pp. 151–167. Springer, Berlin (2011)

    Google Scholar 

  14. Van Leemput P., Rheinländer M., Junk M.: Smooth initialization of lattice Boltzmann schemes. Comput. Math. Appl. 58, 867–882 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vandekerckhove C., Kevrekidis I., Roose D.: An efficient Newton-Krylov implementation of the Constrained Runs scheme for initializing on a slow manifold. J. Sci. Comput. 39, 167–188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gear C.W., Kaper T.J., Kevrekidis I.G., Zagaris A.: Projecting to a slow manifold: singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst. 4, 711–732 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Van Leemput P., Vanroose W., Roose D.: Mesoscale analysis of the equation-free Constrained Runs initialization scheme. Multiscale Model Simul. 6, 1234–1255 (2007)

    Article  MathSciNet  Google Scholar 

  18. Zagaris A., Gear C.W., Kaper T.J., Kevrekidis I.G.: Analysis of the accuracy and convergence of equation-free projection to a slow manifold. ESIAM Math. Model. Numer. Anal. 43, 757–784 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wolf-Gladrow D.A.: Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer, Berlin (2000)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  21. Junk M., Klar A., Luo L.: Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys. 210, 676–704 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Van Leemput, P.: Multiscale and Equation-Free Computing for Lattice Boltzmann Models. PhD thesis, K.U. Leuven (2007)

  23. Cercignani C.: The Boltzmann Equation and its Applications. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  24. Rheinländer M.: A consistent grid coupling method for lattice-Boltzmann schemes. J. Stat. Phys. 121, 49–74 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vandekerckhove, C.: Macroscopic Simulation of Multiscale Systems within the Equation-Free Framework. PhD thesis, K.U. Leuven (2008)

  26. Kelley C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA (1995)

    Book  MATH  Google Scholar 

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Correspondence to Y. Vanderhoydonc.

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Communicated by C. W. Oosterlee and A. Borzi.

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Vanderhoydonc, Y., Vanroose, W. Lifting in hybrid lattice Boltzmann and PDE models. Comput. Visual Sci. 14, 67–78 (2011). https://doi.org/10.1007/s00791-011-0164-6

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  • DOI: https://doi.org/10.1007/s00791-011-0164-6

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