A unified operator splitting approach for multi-scale fluid-particle coupling in the lattice Boltzmann method

Abstract A unified framework to derive discrete time-marching schemes for the coupling of immersed solid and elastic objects to the lattice Boltzmann method is presented. Based on operator splitting for the discrete Boltzmann equation, second-order time-accurate schemes for the immersed boundary method, viscous force coupling and external boundary force are derived. Furthermore, a modified formulation of the external boundary force is introduced that leads to a more accurate no-slip boundary condition. The derivation also reveals that the coupling methods can be cast into a unified form, and that the immersed boundary method can be interpreted as the limit of force coupling for vanishing particle mass. In practice, the ratio between fluid and particle mass determines the strength of the force transfer in the coupling. The integration schemes formally improve the accuracy of first-order algorithms that are commonly employed when coupling immersed objects to a lattice Boltzmann fluid. It is anticipated that they will also lead to superior long-time stability in simulations of complex fluids with multiple scales.

[1]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation , 1993, Journal of Fluid Mechanics.

[2]  J. Broadwell,et al.  Study of rarefied shear flow by the discrete velocity method , 1964, Journal of Fluid Mechanics.

[3]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[4]  Dirk Pflüger,et al.  Lecture Notes in Computational Science and Engineering , 2010 .

[5]  Manfred Krafczyk,et al.  Lattice-Boltzmann Method on Quadtree-Type Grids for Fluid-Structure Interaction , 2006 .

[6]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[7]  Stefano Ubertini,et al.  A coupled lattice Boltzmann-finite element approach for two-dimensional fluid-structure interaction , 2013 .

[8]  T. G. Cowling,et al.  The mathematical theory of non-uniform gases : notes added in 1951 , 1951 .

[9]  L. Luo,et al.  A priori derivation of the lattice Boltzmann equation , 1997 .

[10]  Rick Salmon.  The lattice Boltzmann method as a basis for ocean circulation modeling , 1999 .

[11]  T. Ohwada.  Higher Order Approximation Methods for the Boltzmann Equation , 1998 .

[12]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[13]  P. Español,et al.  Efficient numerical integrators for stochastic models , 2005, physics/0502064.

[14]  M. Suzuki,et al.  Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics , 1985 .

[15]  A. Ladd,et al.  Lattice Boltzmann Simulations of Soft Matter Systems , 2008, 0803.2826.

[16]  L. Einkemmer.  Structure preserving numerical methods for the Vlasov equation , 2016, 1604.02616.

[17]  L. Sirovich,et al.  Modeling a no-slip flow boundary with an external force field , 1993 .

[18]  Paul J. Dellar,et al.  Incompressible limits of lattice Boltzmann equations using multiple relaxation times , 2003 .

[19]  R. Skeel,et al.  Nonlinear Resonance Artifacts in Molecular Dynamics Simulations , 1998 .

[20]  R. Carter.  Lie Groups , 1970, Nature.

[21]  G. Ciccotti,et al.  Algorithms for Brownian dynamics , 2003 .

[22]  Tony Shardlow,et al.  Splitting for Dissipative Particle Dynamics , 2002, SIAM J. Sci. Comput..

[23]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results , 1993, Journal of Fluid Mechanics.

[24]  Paul J. Dellar,et al.  An interpretation and derivation of the lattice Boltzmann method using Strang splitting , 2013, Comput. Math. Appl..

[25]  M. Junk,et al.  Asymptotic analysis of the lattice Boltzmann equation , 2005 .

[26]  S Succi,et al.  Three ways to lattice Boltzmann: a unified time-marching picture. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Shiyi Chen,et al.  A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit , 1998 .

[28]  B. Shi,et al.  Discrete lattice effects on the forcing term in the lattice Boltzmann method. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[30]  Ulf D. Schiller,et al.  Statistical mechanics of the fluctuating lattice Boltzmann equation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  G. D. Byrne,et al.  VODE: a variable-coefficient ODE solver , 1989 .

[32]  Nicos Martys,et al.  Evaluation of the external force term in the discrete Boltzmann equation , 1998 .

[33]  Nawaf Bou-Rabee,et al.  Time Integrators for Molecular Dynamics , 2013, Entropy.

[34]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[35]  Gerhard Wellein,et al.  Comparison of different propagation steps for lattice Boltzmann methods , 2011, Comput. Math. Appl..

[36]  Burkhard Dünweg,et al.  Implicit and explicit solvent models for the simulation of a single polymer chain in solution: Lattice Boltzmann versus Brownian dynamics. , 2009, The Journal of chemical physics.

[37]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[38]  V. Varadarajan.  Lie groups, Lie algebras, and their representations , 1974 .

[39]  G. Quispel,et al.  Acta Numerica 2002: Splitting methods , 2002 .

[40]  M. Lai,et al.  An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity , 2000 .

[41]  Tamar Schlick,et al.  A Family of Symplectic Integrators: Stability, Accuracy, and Molecular Dynamics Applications , 1997, SIAM J. Sci. Comput..

[42]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[43]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[44]  Houman Owhadi,et al.  Long-Run Accuracy of Variational Integrators in the Stochastic Context , 2007, SIAM J. Numer. Anal..

[45]  Alexander N Gorban,et al.  Stability and stabilization of the lattice Boltzmann method. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  P. Ahlrichs,et al.  Simulation of a single polymer chain in solution by combining lattice Boltzmann and molecular dynamics , 1999, cond-mat/9905183.

[47]  H. Trotter.  On the product of semi-groups of operators , 1959 .

[48]  A. Ladd,et al.  Lattice-Boltzmann Simulations of Particle-Fluid Suspensions , 2001 .

[49]  W. Gröbner,et al.  Die Lie-Reihen und ihre Anwendungen , 1960 .

[50]  Rahul Kekre,et al.  Comparison of the static and dynamic properties of a semiflexible polymer using lattice Boltzmann and Brownian-dynamics simulations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  P. Dellar.  Bulk and shear viscosities in lattice Boltzmann equations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  Alexander V. Bobylev,et al.  The error of the splitting scheme for solving evolutionary equations , 2001, Appl. Math. Lett..

[53]  R. Winkler,et al.  Multi-Particle Collision Dynamics -- a Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids , 2008, 0808.2157.

[54]  Gerhard Gompper,et al.  Sedimentation of single red blood cells , 2013 .

[55]  Michel Hénon,et al.  Viscosity of a Lattice Gas , 1987, Complex Syst..

[56]  C. Aidun,et al.  Simulating 3D deformable particle suspensions using lattice Boltzmann method with discrete external boundary force , 2009 .

[57]  B. Alder,et al.  Analysis of the lattice Boltzmann treatment of hydrodynamics , 1993 .

[58]  M. Dupin,et al.  Modeling the flow of dense suspensions of deformable particles in three dimensions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  M. Cates,et al.  Fluctuating lattice Boltzmann , 2004, cond-mat/0402598.

[60]  Ernst Rank,et al.  Two-dimensional simulation of fluid–structure interaction using lattice-Boltzmann methods , 2001 .

[61]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[62]  Li-Shi Luo,et al.  Some recent results on discrete velocity models and ramifications for lattice Boltzmann equation , 2000 .

[63]  Mark E. Tuckerman,et al.  Reversible multiple time scale molecular dynamics , 1992 .

[64]  Kun Xu,et al.  Lattice Boltzmann method and gas-kinetic BGK scheme in the low-Mach number viscous flow simulations , 2003 .

[65]  R. Verberg,et al.  Modeling the interactions between deformable capsules rolling on a compliant surface. , 2006, Soft matter.

[66]  X. Yuan,et al.  Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation , 2006, Journal of Fluid Mechanics.

[67]  Li-Shi Luo,et al.  Unified Theory of Lattice Boltzmann Models for Nonideal Gases , 1998 .

[68]  R. Salmon,et al.  Lattice Boltzmann solutions of the three-dimensional planetary geostrophic equations , 1999 .

[69]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

[70]  X. He,et al.  Discretization of the Velocity Space in the Solution of the Boltzmann Equation , 1997, comp-gas/9712001.

[71]  L. Luo,et al.  Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation , 1997 .

[72]  C. Peskin,et al.  Implicit second-order immersed boundary methods with boundary mass , 2008 .

[73]  E. Atlee Jackson,et al.  Kinetic Models and the Linearized Boltzmann Equation , 1959 .

[74]  A. Malevanets,et al.  Mesoscopic model for solvent dynamics , 1999 .

[75]  Cyrus K. Aidun,et al.  Lattice-Boltzmann Method for Complex Flows , 2010 .

[76]  M. Parrinello,et al.  Accurate sampling using Langevin dynamics. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[77]  Pep Español,et al.  A stochastic Trotter integration scheme for dissipative particle dynamics , 2006, Math. Comput. Simul..

[78]  G. Doolen,et al.  Discrete Boltzmann equation model for nonideal gases , 1998 .