Stability analysis of the implicit finite-difference-based upwind lattice Boltzmann schemes




lattice Boltzmann method, implicit finite-difference schemes, stability


The paper is devoted to the stability analysis of the implicit finite-difference schemes for the system of kinetic equations used for the hydrodynamic computations in the framework of the lattice Boltzmann method. The families of two- and three-layer upwind schemes of the first to fourth approximation orders on spatial variables are considered. An important feature of the presented schemes is that the convective terms are approximated by one finite difference. It is shown that, for the high-order schemes, in the expression for the current viscosity there are no fictitious terms, which makes it possible to perform computations in the whole range of relaxation time values. The stability analysis is based on the application of the von Neumann method to the linear approximations of the schemes. The stability conditions are obtained in the form of inequalities imposed on the Courant number values. It is also shown that the areas of stability domains for the two-layer schemes are greater than for the three-layer schemes in the parameter space. The considered schemes can be used as the fully implicit schemes in computational algorithms directly or in the predictor-corrector methods.

Author Biographies

G.V. Krivovichev

St Petersburg University
• Associate Professor

M.P. Mashchinskaya


  1. S. Succi, The Lattice Boltzmann Equation for Complex States of Flowing Matter (Oxford Univ. Press, Oxford, 2018).
  2. D. A. Bikulov and D. S. Senin, “Implementation of the Lattice Boltzmann Method without Stored Distribution Functions on GPU,” Vychisl. Metody Programm. 14, 370-374 (2013).
  3. D. A. Bikulov, “An Efficient Implementation of the Lattice Boltzmann Method for Hybrid Supercomputers,” Vychisl. Metody Programm. 16, 205-214 (2015).
  4. A. L. Kupershtokh, “Three-Dimensional Simulations of Two-Phase Liquid-Vapor Systems on GPU Using the Lattice Boltzmann Method,” Vychisl. Metody Programm. 13, 130-138 (2012).
  5. A. L. Kupershtokh, D. A. Medvedev, and I. I. Gribanov, “Modeling of Thermal Flows in a Medium with Phase Transitions Using the Lattice Boltzmann Method,” Vychisl. Metody Programm. 15, 317-328 (2014).
  6. J. Wang, Q. Kang, Y. Wang, et al., “Simulation of Gas Flow in Micro-Porous Media with the Regularized Lattice Boltzmann Method,” Fuel 205, 232-246 (2017).
  7. X. He and L.-S. Luo, “A Priori Derivation of the Lattice Boltzmann Equation,” Phys. Rev. E 55 (6), R6333-R6336 (1997).
  8. V. Sofonea and R. F. Sekerka, “Viscosity of Finite Difference Lattice Boltzmann Models,” J. Comput. Phys. 184 (2), 422-434 (2003).
  9. T. Seta and R. Takakashi, “Numerical Stability Analysis of FDLBM,” J. Stat. Phys. 107 (1-2), 557-572 (2002).
  10. X. Shi, X. Huang, Y. Zheng, and T. Ji, “A Hybrid Algorithm of Lattice Boltzmann Method and Finite-Difference-Based Lattice Boltzmann Method for Viscous Flows,” Int. J. Numer. Meth. Fluids 85 (11), 641-661 (2017).
  11. A. Fakhari and T. Lee, “Finite-Difference Lattice Boltzmann Method with a Block-Structured Adaptive-Mesh-Refinement Technique,” Phys. Rev. E 89, 033310-1-033310-12 (2014).
  12. W. Li and W. Li, “A Gas-Kinetic BGK Scheme for the Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows,” Comput. Fluids 162, 126-138 (2018).
  13. L. Chen and L. Schaefer, “Godunov-Type Upwind Flux Schemes of the Two-Dimensional Finite Volume Discrete Boltzmann Method,” Comput. Math. Appl. 75 (9), 3105-3126 (2018).
  14. W. Shao and J. Li, “Three Time Integration Methods for Incompressible Flows with Discontinuous Galerkin lattice Boltzmann method,” Comput. Math. Appl. 75 (11), 4091-4106 (2018).
  15. M. Min and T. Lee, “A Spectral-Element Discontinuous Galerkin Lattice Boltzmann Method for Nearly Compressible Flows,” J. Comput. Phy. 230 (1), 245-259 (2011).
  16. T. Biciusca, A. Horga, and V. Sofonea, “Simulation of Liquid-vapour Phase Separation on GPUs Using Lattice Boltzmann Models with Off-Lattice Velocity Sets,” Comptes Rendus Mécanique 343 (10-11), 580-588 (2015).
  17. G. V. Krivovichev and S. A. Mikheev, “Stability of Three-Layer Finite Difference-Based Lattice Boltzmann Schemes,” Vychisl. Metody Programm. 15, 211-221 (2014).
  18. G. V. Krivovichev and S. A. Mikheev, “Stability Study of Finite-Difference-Based Lattice Boltzmann Schemes with Upwind Differences of High Order Approximation,” Vychisl. Metody Programm. 16, 196-204 (2015).
  19. G. V. Krivovichev and S. A. Mikheev, “On the Stability of Multi-Step Finite-Difference-Based Lattice Boltzmann Schemes,” Int. J. Comput. Meth. 16 (2019).
    doi 10.1142/S0219876218500871
  20. G. V. Krivovichev and E. V. Voskoboinikova, “Application of Predictor-Corrector Finite-Difference-Based Schemes in the Lattice Boltzmann Method,” Vychisl. Metody Programm. 16, 10-17 (2015).
  21. P. Asinari, “Semi-Implicit-Linearized Multiple-Relaxation-Time Formulation of Lattice Boltzmann Schemes for Mixture Modeling,” Phys. Rev. E 63 (2006).
    doi 10.1103/PhysRevE.73.056705
  22. D. R. Rector and M. L. Stewart, “A Semi-Implicit Lattice Method for Simulating Flow,” J. Comput. Phys. 229 (19), 6732-6743 (2010).
  23. N. Cao, S. Chen, S. Jin, and D. Martinez, “Physical Symmetry and Lattice Symmetry in the Lattice Boltzmann Method,” Phys. Rev. E 55 (1), R21-R24 (1997).
  24. M. Bernaschi, S. Succi, and H. Chen, “Accelerated Lattice Boltzmann Schemes for Steady-State Flow Simulations,” J. Sci. Comput. 16 (2), 135-144 (2001).
  25. T. Lee and C.-L. Lin, “An Eulerian Description of the Streaming Process in the Lattice Boltzmann Equation,” J. Comput. Phys. 185 (2), 445-471 (2003).
  26. Y. Li, E. J. LeBoeuf, and P. K. Basu, “Least-Squares Finite-Element Lattice Boltzmann Method,” Phys. Rev. E 69 (2004).
    doi 10.1103/PhysRevE.69.065701
  27. Y. Wang, Y. L. He, T. S. Zhao, et al., “Implicit-Explicit Finite-Difference Lattice Boltzmann Method for Compressible Flows,” Int. J. Mod. Phys. C 18 (12), 1961-1983 (2007).
  28. R.-F. Qiu, Y.-C. You, C.-X. Zhu, and R.-Q. Chen, “Lattice Boltzmann Simulation for High-Speed Compressible Viscous Flows with a Boundary Layer,” Appl. Math. Model. 48, 567-583 (2017).
  29. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems (Wiley, New York, 1967; Mir, Moscow, 1972).
  30. B. S. Garbow, “EISPACK - A Package of Matrix Eigensystem Routines,” Comput. Phys. Commun. 7, 179-184 (1974).
  31. E. A. Prokhorova and G. V. Krivovichev, “Parallel Realization of the Computational Algorithm Based on the Implicit Lattice Boltzmann Equations,” J. Phys. Conf. Ser. 1038 (2018).
    doi 10.1088/1742-6596/1038/1/012041



How to Cite

Кривовичев Г.В., Мащинская М.П. Stability Analysis of the Implicit Finite-Difference-Based Upwind Lattice Boltzmann Schemes // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2019. 20. 116-127. doi 10.26089/NumMet.v20r212



Section 1. Numerical methods and applications

Most read articles by the same author(s)

1 2 > >>